Review of Local and Global Existence Results for Stochastic Pdes with Lévy Noise

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020241 DYNAMICAL SYSTEMS Volume 40, Number 10, October 2020 pp. 5639–5710

REVIEW OF LOCAL AND GLOBAL EXISTENCE RESULTS FOR STOCHASTIC PDES WITH LÉVY NOISE

Justin Cyr1, Phuong Nguyen1,2, Sisi Tang1 and Roger Temam1,∗ 1 Department of Mathematics Indiana University Swain East Bloomington, IN 47405, USA 2 Department of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, USA

(Communicated by Nikolay Tzvetkov)

Abstract. This article is a review of Lévy processes, stochastic integration and existence results for stochastic differential equations and stochastic partial differential equations driven by Lévy noise. An abstract PDE of the typical type encountered in fluid mechanics is considered in a stochastic setting driven by a general Lévy noise. Existence and uniqueness of a local pathwise solution is established as a demonstration of general techniques in the area.

1. Introduction. In this article we review results of existence and uniqueness of local pathwise solutions to stochastic partial differential equations (SPDEs) driven by a Lévy noise. Lévy processes are canonical generalizations of Wiener processes that possess jump discontinuities. Lévy noise is fitting in stochastic models of fluid dynamics as a way to represent interactions that occur as abrupt jolts at random times, such as bursts of wind. Models based on SPDEs with Lévy noise have been used to describe observational records in paleoclimatology in [14], with the jump events of the Lévy noise proposed as representing abrupt triggers for shifts between glacial and warm climate states. The articles [40] and [24] use processes with jumps to model phase transitions in bursts of wind that contribute to the dynamics of El Niño. In addition, [36] uses processes with jumps to model phase transitions in precipitation dynamics. See [31] for a comparison of Wiener noise and Lévy noise in modeling climate phenomena. As the body of work and interest in SPDEs with Lévy noise is growing, see e.g., [5], [6], [7], [9], [28], [29]; see also the book [13] and the book [32] by Peszat and Zabczyk, we feel it is useful to present foundational existence results for SPDEs with Lévy noise along with more specialized techniques that have been used in fluid dynamics together in a synthetic form. As the backdrop of this review we establish existence and uniqueness of local pathwise solutions to an SPDE of a typical form encountered in fluid dynamics

2020 Mathematics Subject Classiﬁcation. Primary: 35R60, 60H15; Secondary: 60G51, 60H10. Key words and phrases. Stochastic partial diﬀerential equations, Lévy noise, pathwise solutions, Galerkin approximation, Poisson random measure. The authors are supported by NSF grant DMS-1510249. ∗ Corresponding author.

5639 5640 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM driven by a Lévy noise. The equation is written in the abstract form  du + [Au + B(u, u) + F (u)] dt = G(u) dW (t) + R K(u(t−), ξ) dπ(t, ξ)  E0 b  R + E\E K(u(t−), ξ) dπ(t, ξ), (1)  0 u(0) = u0, where u takes values in a real, separable Hilbert space H. On the left-hand side of (1), A is a generally unbounded linear operator on H, B is a bilinear form of the type often encountered in fluid dynamics and F is a locally Lipschitz function. The terms on the right-hand side of (1) are stochastic integrals that represent the effects of a Lévy noise. First on the right-hand side of (1) is stochastic integration with respect to a Wiener process. Second and third on the right-hand side of (1) are the discontinuous parts of the Lévy noise, representing the bounded and unbounded jumps, respectively. The abstract setting of equation (1) includes long-studied examples in fluid mechanics such as the Navier-Stokes equations in 2D and the primitive equations of the atmosphere and the oceans, as in e.g [12]; see also [29] – although we do not treat the primitive equations explicitly in this article. We adopt an abstract framework as a vehicle for presenting general arguments that, we hope, can be modified to settings outside the scope of equation (1) presented here. As an extension of the scope presented here, a subset of the authors consider in [37], [30], the case where F is a more general monotone operator than that considered here in Part2. While solving equation (1) serves as the backdrop of this article, particular attention is given to a detailed treatment of well-known global and local existence and uniqueness results for stochastic differential equations with Lévy noise in finite dimensions. We include these well-known results in Part1 of the article for completeness in our treatment of equation (1) in Part2 and also to demonstrate natural techniques for establishing local and global existence. The foundational existence results covered in Part1 of the article will be invoked in Part2 to establish existence and uniqueness of approximate solutions (1) in a Galerkin approximation scheme. We believe that a presentation of well-known and practical existence results along with an application in solving an SPDE such as (1) could act as a bridge into the subject of SPDEs with Lévy noise for experts in PDEs. The treatment of equation (1) in this article is similar in several ways to the analysis in the Wiener noise case, i.e., equation (1) with K = K = 0, which was carried out in [12]. As usual we cut off the nonlinear term B. As in [12] we choose to truncate the distance, in a certain space, between u and the solution u∗ to the linearized equation ( du + Au dt = 0, ∗ ∗ (2) u∗(0) = u0. This type of truncation allows local solutions to equation (1) to exist up to a stopping time that is positive a.s., regardless of the size of the initial data u0, using the fact that u and u∗ agree at time t = 0. The overall strategy of the Galerkin scheme is analogous to [12]. We establish uniform bounds on the Galerkin approximations in the same spaces, obtain tightness of the laws of the Galerkin approximations and use the Skorohod coupling theorem to pass to the limit and obtain a martingale solution to a truncated version of equation (1). However, the implementation of these steps is necessarily different from the Wiener noise case considered in [12] because of the discontinuity of the Lévy noise. A classical compactness result used in the REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5641

Wiener noise case that relies on continuity in time is replaced by a probabilistic compactness result. In a way similar to [7], [9] and [29] we establish tightness of the laws of the Galerkin approximations using a suﬃcient condition proved by Aldous in [1] for tightness of measures on the space of càdlàg functions endowed with the Skorohod topology. Also as in [7], [9] and [29] we apply a modiﬁed version of the Skorohod coupling theorem to facilitate the passage to the limit in the discontinuous stochastic integral terms on the right-hand side of (1). The article is organized into Parts1 and2, with background for the stochastic setting reviewed in Part1 and existence and uniqueness for equation (1) established in Part2. We begin Part1 in Section2 by collecting background information on Lévy processes and the associated notions of stochastic integration, which appear on the right-hand side of equation (1); at times we refer here to [8]. In Section3 we consider an SDE of the form

( dY = F (s, Y (s−)) ds + G(s, Y (s−)) dW (s) + R K(s, Y (s−), ξ) dπ(s, ξ), E0 b Y (0) = Y0, (3) which, in the absence of a stochastic integral with respect to dπ, is driven by a Lévy process that possesses only jumps of small (i.e., bounded) size. Lévy noise with unbounded jumps may be incorporated once existence with bounded jumps is established using a standard argument known as piecing-out, or interlacing, the jumps. Indeed, in our approach to solving equation (1) the unbounded jumps of the Lévy noise are absent until the piecing-out argument is presented at the very end of the article. We begin in Subsection 3.1 by imposing global Lipschitz and linear growth conditions on F , G and K and establishing global existence and uniqueness of a solution Y , as well as continuity on initial data. In Subsection 3.2 we establish existence and uniqueness of a local solution to (3), i.e., a solution up to a stopping time τ that is positive a.s., under local Lipschitz and linear growth conditions on F , G and K. The results of Section3 will serve as a main tool in implementing a Galerkin scheme for solving equation (1). Part2, where the focus of the article turns to equation (1), begins in Section 4. We state the precise assumptions that will be imposed on the terms A, B, F , G, K and K that appear in (1) in Subsection 4.1. The main result, which is existence and uniqueness of a local pathwise solution to the abstract equation (1), is stated as Theorem 4.3 in Subsection 4.2 after the precise notion of pathwise solution is defined. In Subsection 4.3 we introduce truncated versions of equation (1), with K = 0 among other modifications, that will be solved as steps toward establishing the main result. We also define martingale solutions, a weaker type of solution than pathwise that will be obtained as an intermediate step. In Section5 we n ∞ define Galerkin approximations (u )n=1 to the truncated equation. We establish n ∞ bounds, that are independent of n, on the Galerkin approximations (u )n=1 in Subsection 5.1. We verify the Aldous condition and establish tightness of the laws n ∞ of (u )n=1 in Section6. This yields the law (i.e., probability distribution) of a candidate solution to the truncated version of (1) through weak convergence of the n ∞ laws of (u )n=1 along a subsequence. We pass to the limit in Section7 by applying a modified version of the Skorohod coupling theorem and obtain existence of a global martingale solution to the truncated version of (1). In Section8 we establish pathwise uniqueness for the truncated version of (1), which leads to existence and uniqueness of a global pathwise solution due to an infinite-dimensional version of the 5642 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Yamada-Watanabe theorem established by Gyöngy and Krylov in [19]. In Section 9 we remove the truncation on the B term in equation (1), still with K = 0, and obtain local existence of a pathwise solution. Concrete examples of SPDEs covered by the abstract framework of equation (1) are given in Subsection 9.2. In Section 10 we incorporate the K term into (1) by applying the well-known piecing-out argument.

Part 1. Foundations 2. Lévy processes and stochastic integration. In this section we recall Lévy processes and the associated notions of stochastic integration. We focus on Wiener processes and Poisson random measures. 2.1. Stochastic setting. Let H and U be two separable Hilbert spaces, possibly infinite dimensional. Let (Ω, F, P) be a probability space. A collection of σ-fields (Ft)t≥0 is called a filtration if Fs ⊂ Ft ⊂ F, ∀s ≤ t. We call (Ω, F, (Ft)t≥0, P) a filtered probability space. We also assume that the filtered probability space satisfies the usual conditions, i.e., (Ω, F0, P) is complete, and Ft = ∩s≥tFs for every t ≥ 0, which means that the filtration is right-continuous. We recall below the main stochastic processes that drive the noise in the SPDEs that we consider. Each stochastic process (X(t))t≥0 that we consider is assumed to be defined on the same filtered probability space (Ω, F, (Ft)t≥0, P) and (X(t))t≥0 is assumed to be adapted, meaning that the random variable X(t) is Ft-measurable for every t ≥ 0. Definition 2.1. We say that a U-valued stochastic process (W (t), t ≥ 0) defined on (Ω, F, P) is a Wiener process if i) W (0) = 0 a.s., ii) W has independent increments, i.e., for 0 ≤ s ≤ t, W (t) − W (s) is independent of σ(W (u), u ≤ s), iii) W has stationary increments, i.e., W (t) − W (s) =D W (t − s), ∀t > s ≥ 0, and iv) W is stochastically continuous, i.e., ∀ > 0,

lim P(|W (t)|U > ε) = 0, t↓0 v) W has continuous paths a.s., i.e., with probability 1 the function t 7→ W (t) is continuous from [0, ∞) to U.

A U-valued stochastic process (L(t))t≥0 that is defined on (Ω, F, P) and adapted to (Ft)t≥0 and satisfies conditions (i)-(iv) is called a Lévy process. Hence, the only (and major) difference between Lévy processes and Wiener processes is the possible occurrence of discontinuities in Lévy processes. Remember however that every Lévy process has a càdlàg modification. It can be shown that a U-valued Wiener process W has the property that for all t1, t2, . . . , tn ≥ 0 and y1, y2, . . . , yn ∈ U, the random vector

(hW (t1), y1iU , hW (t2), y2iU ,..., hW (tn), yniU ) has a mean zero multivariate normal distribution on Rn with covariance matrix Σ given by

Σi,j = E[hW (ti), yiiU hW (tj), yjiU ], see, e.g., Theorem 4.20 in [32]. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5643

Next, we define Poisson random measures and Poisson point processes. These concepts are used to represent noise with respect to discontinuous Lévy processes in our SPDE (see [8]). Definitions of these concepts can be found in many stochastic calculus or SPDE books. Here, we follow the definitions in [21] Chapter 1, Sections 8 and 9; see also [8]. Let (E, E) be a measurable space. Let N := {0, 1, 2,...} denote the set of nonnegative integers and let N := N ∪ {∞}. Let M be the collection of non-negative, possibly infinite, N-valued measures on (E, E), and BM be the smallest σ-field on M such that the maps from M to N given by µ 7→ µ(Γ) are measurable for every Γ ∈ E. Definition 2.2. Let λ be a σ-finite measure on (E, E).A Poisson random measure on E with intensity measure λ is a (M, BM)-valued random variable π defined on the probability space (Ω, F, P) with the property that for all pairwise disjoint sets Γ1,..., Γn ∈ E the N-valued random variables π(Γ1), . . . , π(Γn) are independent Poisson random variables with parameters λ(Γ1), . . . , λ(Γn), i.e.,

k P[π(Γi) = k] = λ(Γi) exp(−λ(Γi))/k!, k = 0, 1, 2, . . . , i = 1, 2, . . . , n.

It is possible here that λ(Γi) = ∞. In this case we use the convention that a Poisson random variable with parameter ∞ is identically equal to infinity. We call πb := π − λ the compensated Poisson random measure. We will use notation b to mean “compensated”. Note that for A ∈ E, the real-valued random variable πb(A) is well-defined when λ(A) < ∞. It can be shown that given any σ-finite measure λ on (E, E), there exists a Poisson random measure π with intensity λ. See [21] Theorem 8.1 for a proof of this fact. Let (E, E) be a measurable space. A point function is a mapping p: Dp → E whose domain Dp is a countable subset of (0, ∞). Each point function p defines an N-valued measure Np( dt, dx) on (0, ∞) × E by

Np((0, t] × Γ) := #{s ∈ Dp : s ∈ (0, t], p(s) ∈ Γ}, t > 0, Γ ∈ E. (4)

Let ΠE be the collection of point functions on E and B(ΠE) be the smallest σ-ﬁeld on ΠE such that the functions of the form p 7→ Np((0, t] × Γ) are measurable for all t > 0 and all Γ ∈ E.

Deﬁnition 2.3. A Poisson point process Ξ on (E, E) is a (ΠE, B(ΠE))-valued random variable on (Ω, F, P) such that NΞ( dt, dx) is a Poisson random measure on (0, ∞) × E. A Poisson point process Ξ is said to be stationary if and only if the intensity measure λ of its Poisson random measure NΞ is of the form dλ = dt ⊗ dν for some σ-ﬁnite positive measure ν on (E, E). In this case, for every set Γ ∈ E the N-valued process (NΞ((0, t] × Γ))t≥0 is a Poisson process with rate ν(Γ). In the setting of SPDEs, we are only interested in stationary Poisson point processes. In the setting of SPDE, we will assume that the Lévy noise π is induced by a stationary Poisson point process Ξ on a measurable space (E, E). According to (4), we can see that X π := NΞ = δ(s,Ξ(s)). (5)

s∈DΞ In Section5 we will make use of the Burkholder-Davis-Gundy inequality (Theo- rem 2.5). Before stating this inequality we recall the notion of stopping time. 5644 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Definition 2.4. Let (Ft)t≥0 be a filtration on a probability space (Ω, F, P).A nonnegative random variable τ :Ω → [0, ∞] is called an Ft-stopping time (or just stopping time when the filtration is clear) if

{τ ≤ t} ∈ Ft, for all t ≥ 0.

Theorem 2.5. Let H be a real, separable Hilbert space. Let (Ω, F, (Ft)t≥0, P) be a ﬁltered probability space. For every 1 ≤ p < ∞ there exists a constant Cp > 0 depending only on p, such that for every càdlàg H-valued Ft-martingale M such 2 that E|M(t)|H < ∞ for every t ≥ 0 and M(0) = 0, and for every Ft-stopping time τ one has p p/2 E sup |M(t)|H ≤ CpE[M]τ , (6) t∈[0,τ] where [M] is the quadratic variation of M.

For a proof see, e.g., [25]. The fact that the constant Cp does not depend on the martingale M or the stopping time τ is a key part of the conclusion. Since the constant Cp does not depend on M or τ, one can employ the BDG inequality when making a priori estimates for stochastic partial differential equations. See, e.g., [27] for the definition of quadratic variation of a Hilbert space-valued martingale and a general treatment of the subject. For the purpose of treating SPDEs we are only interested in applying inequality (6) when M is a process formed by stochastic integration with respect to a Wiener process or compensated Poisson random measure, both of which will be defined later in this section. Instead of giving a detailed discussion on quadratic variation, we will be content with focusing on the two cases that are of interest to us. We give the quadratic variation of a stochastic integral with respect to a Wiener process in Example1 and state the BDG inequality in this case. We give the quadratic variation of a stochastic integral with respect to a compensated Poisson random measure in Proposition2. 2.2. Stochastic integration w.r.t. square-integrable Lévy martingales. In this section we will define stochastic integration with respect to square-integrable R T martingales by a limiting procedure. This will be applied to the 0 G(u) dW term in the SPDE. We refer to the definitions in Chapter 22 of [22]. In this section, we still work on the probability space (Ω, F, (Ft)t≥0, P) with usual assumptions.

Definition 2.6. Let (Ω, F, (Ft)t≥0, P) be a filtered probability space with usual assumptions. The σ-field on the product space Ω × [0,T ] generated by all left- continuous, adapted, real-valued processes is called the predictable σ-field and it is denoted by P[0,T ]. A stochastic process is said to be predictable process if it is P[0,T ]-measurable.

There are several equivalent ways of generating the predictable σ-ﬁeld P[0,T ]. We describe them in a lemma below, which can be found in Chapter 22 of [22]. Item (i) in Lemma 2.7 gives a hint at why we work with predictable processes.

Lemma 2.7. The predictable σ-ﬁeld P[0,T ] is the σ-ﬁeld generated by each of the following:

i) F0 × [0,T ], and the sets A × (t, T ] with A ∈ Ft, 0 ≤ t ≤ T , ii) all left continuous, adapted, real-valued processes, iii) all continuous, adapted, real-valued processes. The following measurability property plays a crucial role in the construction of the stochastic integral in this section. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5645

Deﬁnition 2.8. Let L be a Lévy process on a ﬁltered probability space (Ω, F, (Ft)t≥0, P) taking values in a real, separable Hilbert space U. We say that L is an Ft-Lévy process if L is adapted to (Ft)t≥0, and

L(t) − L(s) is independent of Fs for all t ≥ s ≥ 0. (7)

We will say that W is an Ft-Wiener process if and only if W is both a Wiener process and an Ft-Lévy process. We will use the term Ft-Poisson process in the same manner. We say that a Poisson point process Ξ on a measurable space (E, E) is an Ft-Poisson point process if (NΞ((0, t] × Γ))t≥0 is an Ft-Poisson process for every Γ ∈ E, i.e., if

NΞ((0, t] × Γ) − NΞ((0, s] × Γ) is independent of Fs for all t ≥ s ≥ 0. Remark 1. We mention that condition (7) is stronger than the condition that a Lévy process has independent increments. The independence of increments condition implies that every Lévy process L satisfies condition (7) with respect to its natural filtration Fet := σ(L(s): s ≤ t). However, the filtration (Fet)t≥0 may not be appropriate for treating SPDEs because the usual conditions, i.e., completeness and right-continuity, may not be satisfied. Despite this, the filtration (Fet)t≥0 can always be enlarged to a filtration (Ft)t≥0 that is complete and right-continuous and in such a way that L is an Ft-Lévy process (see Proposition 2.1.13 of [34]). So, one should think of the filtration (Ft)t≥0 in condition (7) as satisfying the usual conditions but possibly being larger than the natural filtration (Fet)t≥0 generated by L.

In the remainder of Section 2.2 we work with a process M on (Ω, F, (Ft)t≥0, P) taking values in a real, separable Hilbert space U that satisﬁes the following assumption: The stochastic process M is a U-valued, square-integrable, mean-zero Ft-Lévy process. For example, every Ft-Wiener process satisﬁes these conditions. Theorem 2.9. Let M satisfy the above assumptions. Then the following statements hold: i) There exists a bounded, linear, symmetric, positive operator Q: U → U such that

E (hM(t), xiU hM(s), yiU ) = (t ∧ s) hQx, yiU for all t, s ≥ 0, for all x, y ∈ U. (8) ii) Furthermore, Q is of trace class and

E hM(t),SM(t)iU = tTr(SQ), for all S ∈ L(U, U), 2 and in particular E|M(t)|U = tTrQ. We denote the space of bounded linear operators from a Hilbert space U to a Hilbert space H by L(U, H). See Theorem 4.44 in [32] for a proof of Theorem 2.9, which is valid even when M is just a square-integrable Lévy martingale. Definition 2.10. Let M be as before. The positive, trace class operator Q defined + by (8) is called the covariance operator of M. We write L1 (U) for the space of all + positive, trace class operators on U. So, Theorem 2.9 says that Q ∈ L1 (U). Recall that H,U are two Hilbert spaces. The processes that we will integrate with respect to M will be L(U, H)-valued. We begin by defining the stochastic integral of certain step functions called simple processes. 5646 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Definition 2.11. We denote by S(U, H) the space of all L(U, H)-valued stochastic processes Ψ of the form m−1 X Ψ(ω, s) = χAj (ω)χ(tj ,tj+1](s)Φj (9) j=0 where 0 = t0 < t1 < ··· < tm, Aj ∈ Ftj , and Φj ∈ L(U, H). The elements of S(U, H) are called simple processes. Remark 2. We emphasize that the space S(U, H) of simple processes depends on the filtration (Ft)t≥0 through the assumption that Aj ∈ Ftj . This condition means that simple processes Ψ ∈ S(U, H) are predictable in the sense of Definition 2.6. Predictability of simple processes is crucial in the proof of the Itô isometric formula in Proposition1, which is what allows us to extend the notion of stochastic integral beyond simple processes. The predictability assumption is crucial for developing the entire theory of stochastic integration. Definition 2.12. For a simple process Ψ ∈ S(U, H), we define the stochastic integral of Ψ with respect to M by Z t m−1 M X Ψ(s) dM(s) := It (Ψ) := χAj Φj(M(tj+1 ∧ t) − M(tj ∧ t)) for all t ≥ 0. 0 j=0 (10) M Thus, the stochastic integral It (Ψ) is an H-valued stochastic process. One more piece of notation is needed to state the isometric formula.

Notation. Let L2(U, H) denote the space of Hilbert-Schmidt operators from U to H equipped with the Hilbert-Schmidt norm ∞ 2 X kSk := |Su |2 , L2(U,H) n H n=1 ∞ where (un)n=1 is an orthonormal basis of U. This norm makes L2(U, H) a Hilbert space and the norm does not depend on the choice of the orthonormal basis of U. The basic isometric formula, also called the Itô isometry, is given below. Proposition 1 (Itô isometry for simply process). For every Ψ ∈ S(U, H) and t ≥ 0 we have Z t 2 M 2 1/2 E|It (Ψ)|H = E Ψ(s)Q ds, (11) 0 L2(U,H) where Q is the covariance operator of M. Note that the right-hand side of (11) is ﬁnite because 2 1/2 ΦQ ≤ kΦkL(U,H) · Tr(Q) < ∞ L2(U,H) for every Φ ∈ L(U, H). The independence condition (7) in Deﬁnition 2.8 plays a crucial role in the proof of Proposition1. See Proposition 8.6 in [32] for a proof of Proposition1. M Next we would like to extend the stochastic integration map IT to a larger space of integrands that contains the simple processes. Before that we must construct the completion (or closure) of S(U, H). REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5647

+ Deﬁnition 2.13. Let Q ∈ L1 (U). We deﬁne an inner product on the space U0 := Q1/2(U) by

D −1/2 −1/2 E 1/2 hx, yiU := Q x, Q y for all x, y ∈ Q (U), (12) 0 U where Q−1/2 : Q1/2(U) → N (Q1/2)⊥ is the pseudoinverse of Q1/2. The restric- 1/2 −1/2 tion Q |N (Q1/2)⊥ is a bijection onto its image and Q is defined to be the inverse mapping. Although Q1/2(U) is typically not a closed subspace of U, the 1/2 1/2 ⊥ map Q |N (Q1/2)⊥ : N (Q ) → U0 is an isometric isomorphism under the inner 1/2 product in (12), so U0 is complete. We will occasionally write Q (U) instead of 1/2 U0 for the range of Q endowed with the inner product in (12) in order to make the dependence on the operator Q explicit in the notation. + If Q ∈ L1 (U), then Q is a symmetric compact operator, so there exists an ∞ orthonormal basis (un)n=1 of U consisting of eigenvectors of Q. It is easy to see 1/2 ∞ that the nonzero terms in Q un n=1 form an orthonormal basis for U0. Therefore, the Itô isometry in (11) can be restated in the equivalent form Z t E|IM (Ψ)|2 = E kΨ(s)k2 ds, for all Ψ ∈ S(U, H), t ≥ 0. (13) t H L2(U0,H) 0 The right-hand side of (13) is the norm squared in the space 2 X := L (Ω × [0,T ], F ⊗ B([0,T ]), dP ⊗ dt; L2(U0,H)) . Here and below B([0,T ]) denotes the Borel σ-field on [0,T ]. We observed above that every Ψ ∈ S(U, H) belongs to the space X . We will continue to use S(U, H) to denote the space of equivalence classes of simple processes in the space X . Note that if Ψ, Φ ∈ S(U, H) and Ψ = Φ in the space X , then it does not necessarily follow that Ψ and Φ are equal in L(U, H), dP dt-a.e. Instead, Ψ = Φ in X means only that ΨQ1/2 = ΦQ1/2, dP dt-a.e. That is, Ψ and Φ do not necessarily agree on all of U, but they do agree on the range of Q1/2, dP dt-a.e. Equation (13) shows M that It is well-defined on the space S(U, H) viewed as equivalence classes in X , 1/2 1/2 M M i.e., if Ψ, Φ ∈ S(U, H) and ΨQ = ΦQ , dP dt-a.e., then It (Ψ) = It (Φ) in 2 M 2 L (Ω; H). We can now extend It : S(U, H) → L (Ω; H) uniquely to an isometry on the closure of S(U, H) in the space X . The resulting isometry is the stochastic integral with respect to M. Before stating the general properties of the stochastic integral with respect to M we pause to identify the closure of S(U, H) in the space X ; see Lemma 8.13 in [32] for a proof. Lemma 2.14. The closure of S(U, H) in the space X is the subspace of predictable processes in X . In other words, S(U, H) is dense in the space L2 (H) := L2(Ω × [0,T ], P , dP ⊗ dt; L (U ,H)), (14) U0,T [0,T ] 2 0 where P[0,T ] is the σ-field of predictable sets (see Definition 2.6). Note that the space L2 (H) of integrands for stochastic integration with respect U0,T to M depends on the filtration (Ft)t≥0 through the requirement of predictability. We gather the main facts about the stochastic integral with respect to M below. Theorem 2.15. i) For every t ∈ [0,T ], IM : L2 (H) → L2(Ω; H) is an isome- t U0,t try, i.e., Z t E IM (Ψ),IM (Φ) = E hΨ(s), Φ(s)i ds (15) t t H L2(U0,H) 0 5648 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

and E|IM (Ψ)|2 = E R t kΨ(s)k2 ds for all Ψ, Φ ∈ L2 (H). t H 0 L2(U0,H) U0,t ii) For every Ψ ∈ L2 (H) the process IM (Ψ) is a square-integrable H- U0,T t t∈[0,T ] valued martingale that begins at 0. iii) For every Ψ ∈ L2 (H) the angle bracket of IM (Ψ) is given by the U0,T t t∈[0,T ] formula Z t IM (Ψ) = kΨ(s)k2 ds. (16) t L2(U0,H) 0 iv) Let A ∈ L(H,V ) where V is a real, separable Hilbert space. For every Ψ ∈ L2 (H) we have AΨ ∈ L2 (V ) and AIM (Ψ) = IM (AΨ). That is, bounded U0,T U0,T t t operators can be passed inside the stochastic integral.

M Proof. Statement i) follows from the construction of It via Proposition1. The remaining statements hold for simple processes and extend to the case of Ψ ∈ L2 (H) because S(U, H) is dense in L2 (H). U0,T U0,T Example 1. Let W be a U-valued Wiener process. Then W is square-integrable, + so it has a covariance operator Q ∈ L1 (U). In addition, W is an Ft-Wiener process with respect to its natural ﬁltration. We denote the space of integrands for stochastic integration with respect to W by L2 (H). Since W is continuous U0,T R t a.s., 0 Ψ(s) dW (s) is continuous a.s. when Ψ ∈ S(U, H). In fact, the square- W integrable H-valued martingale It (Ψ) t∈[0,T ] is continuous a.s. for every integrand Ψ ∈ L2 (H); see [34]. In particular, the quadratic variation of IW (Ψ) is equal to U0,T its angle bracket. So (16) gives Z t [IW (Ψ)] = kΨ(s)k2 ds. (17) t L2(U0,H) 0 The upper bound in the BDG inequality for stochastic integrals with respect to W takes the following form: for every 1 ≤ p < ∞ there exists a constant Cp ∈ (0, ∞) such that for every F -stopping time τ and every Ψ ∈ L2 (H) we have t U0,T

Z t p Z τ p/2 2 E sup Ψ(s) dW (s) ≤ CpE kΨ(s)k ds . (18) L2(U0,H) t∈[0,τ] 0 H 0 2.3. Integration with respect to Poisson random measures. In this section we review stochastic integration with respect to Poisson ransom measures that arise from stationary Poisson point processes. Let Ξ be a stationary Ft-Poisson point process on E with intensity measure ν and let H be a separable, real Hilbert space. Then, π := NΞ is the corresponding stationary Poisson random measure as deﬁned in (5). We consider the following spaces of functions for integration with respect to the Poisson random measure π induced by Ξ. For q ∈ [1, ∞] we introduce the notation

q q Fν,T (H) := L (Ω × [0,T ] × E, P[0,T ] ⊗ E, dP ⊗ dt ⊗ dν; H). (19) For the purpose of stochastic integration we will only be interested in the spaces 1 2 Fν,T (H) and Fν,T (H). For each fixed ω ∈ Ω, π(ω) is a deterministic N-valued 1 R R measure on (0, ∞) × E. For each f ∈ Fν,T (H), the stochastic integral (0,t] E f dπ is defined in the obvious way — as the Bochner integral of fω with respect to the 1 measure π(ω) for each fixed ω ∈ Ω. In addition, for each f ∈ Fν,T (H) the integral REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5649

R t R f(s, ξ) dν(ξ) ds exists P-a.s. and is Ft-measurable as an H-valued random 0 E0 variable. In this case we deﬁne Z t Z Z t Z Z t Z f(s, ξ) dπb(s, ξ) := f(s, ξ) dπ(s, ξ) − f(s, ξ) ds dν(ξ), (20) 0 E0 0 E0 0 E0 for each t ∈ [0,T ]. In Theorem 2.16 below, the stochastic integral with respect to the 1 compensated Poisson random measure πb in (20) will be extended from Fν,T (H) ∩ 2 2 Fν,T (H) to all of the space Fν,T (H). See, e.g., page 62 in [21] or page 297 in [6] for a proof of Theorem 2.16.

Theorem 2.16. Let Ξ be a stationary Ft-Poisson point process on a measurable space (E, E) with intensity measure ν and let π denote the corresponding Poisson random measure. Then the following statements hold. 1 1 i) (Integrands in Fν,T (H)) Let f ∈ Fν,T (H). Then the following statements hold. R R R t R a) E (0,t] E |f(s, ξ)|H dπ(s, ξ) = E 0 E |f(s, ξ)|H dν(ξ) ds < ∞ for every t ∈ [0,T ]. R R b) For each t ∈ [0,T ] the H-valued integral (0,t] E f(s, ξ) dπ(s, ξ) exists a.s. and is equal to the absolutely convergent sum P f(s, Ξ(s)). s∈(0,t]∩DΞ c) For each t ∈ [0,T ] we have Z Z Z t Z E f(s, ξ) dπ(s, ξ) = E f(s, ξ) dν(ξ) ds. (0,t] E 0 E

2 ii) (Integrands in Fν,T (H)) 1 2 R R a) For f ∈ Fν,T (H) ∩ Fν,T (H), the process f(s, ξ) dπ(s, ξ) is (0,t] E b t∈[0,T ] a square-integrable H-valued martingale on [0,T ] and

Z Z 2 Z t Z 2 E f(s, ξ) dπb(s, ξ) = E |f(s, ξ)|H dν(ξ) ds. (21) (0,t] E H 0 E

1 2 2 b) Fν,T (H) ∩ Fν,T (H) is dense in Fν,T (H). 2 ∞ 1 2 c) Given f ∈ Fν,T (H), let (fn)n=1 be a sequence in Fν,T (H) ∩ Fν,T (H) that ∞ 2 R R converges to f in Fν,T (H). By iia) the sequence fn dπ is (0,t] E b n=1 Cauchy in the space of square-integrable H-valued martingales. Further- ∞ more, the limit does not depend on the particular sequence (fn) . There- n=1 fore, we can deﬁne the H-valued process R R f(s, ξ) dπ(s, ξ) to (0,t] E b t∈[0,T ] be the limit of any such sequence. By construction, this is a square-integrable H-valued martingale and equation (21) continues to hold.

πb 2 2 Deﬁnition 2.17. For T > 0 we deﬁne a map IT : Fν,T (H) → L (Ω; H) by Z Z πb IT (f) := f(s, ξ) dπb(s, ξ). (22) (0,T ] E

πb Equation (21) says that IT is an isometry. Below we recall the quadratic variation and BDG inequality for stochastic integrals with respect to the compensated Poisson random measure π. We refer to [8] for details. 5650 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Theorem 2.18. Let Ξ be a stationary Ft-Poisson point process on a measurable space (E, E) with intensity measure ν and let π denote the associated Poisson ran- 2 dom measure. For every f ∈ Fν,T (H) the quadratic variation of the H-valued πb martingale It (f) t∈[0,T ] is given by Z Z πb 2 [I (f)]t = |f(s, ξ)|H dπ(s, ξ). (23) (0,t] E See Theorem 8.23 in [32] for a proof. While the proof of Theorem 8.23 in [32] is written for real-valued integrands instead of H-valued integrands it is easy to see that the proof remains valid for H-valued integrands when products of real numbers are replaced by inner products in H. With (23) in hand the upper bound in the BDG inequality from Theorem 2.5 takes the following form for stochastic integrals with respect to the compensated Poisson random measure πb.

Proposition 2. Let Ξ be a stationary Ft-Poisson point process on a measurable space (E, E) with intensity measure ν and let π denote the associated Poisson random measure. For every 1 ≤ p < ∞ there exists a constant Cp ∈ (0, ∞) such that 2 for every Ft-stopping time τ and for every f ∈ Fν,T (H) we have Z Z p Z Z p/2 2 E sup f(s, ξ) dπb(s, ξ) ≤ CpE |f(s, ξ)|H dπ(s, ξ) . (24) t∈[0,τ] (0,t] E H (0,τ] E

We emphasize that the constant Cp in inequality (24) does not depend on the function f or the stopping time τ. We will use this inequality in Section5 to make a priori estimates for an SPDE with Lévy noise. We now recall Itô’s formula in order to apply real-valued functions to solutions of SDEs. We refer to Theorem 27.2 in [26] for a proof of the Itô formula in the general context of Hilbert space-valued semimartingales and we refer to [8] for a statement of the Itô formula in the special case of solutions to SDEs with Lévy noise. Before stating the Itô formula we lay out the setting for the SDE in the statement of the Itô formula. Let (Ω, F, (Ft)t≥0, P) be a filtered probability space. Let W be an 1/2 Ft-Wiener process. Let Q be the covariance operator of W and let U0 := Q (U) as in Definition 2.13. Let π be a Poisson random measure on a measurable space (E, E) that is induced by an Ft-Poisson point process that is independent of W . The intensity measure of π has the form dt ⊗ dν for some σ-finite measure ν on (E, E). We will state the Itô formula for processes of the form Z t Z t Z Z X(t) := X0 + F (s) ds + Ψ(s) dW (s) + f(s, ξ) dπb(s, ξ), (25) 0 0 (0,t] E 2 1 where X0 ∈ L (Ω, F0, P; H), F :Ω × [0,T ] → H satisfies F ∈ L ([0,T ]; H) a.s., Ψ ∈ L2 (H) and f ∈ F2 (H) for some T > 0. U0,T ν,T Theorem 2.19. Let X be as in (25). Let ψ : H → R be a C2 function such that D2ψ is uniformly continuous on bounded subsets of H as a mapping into the space of Hilbert-Schmidt operators on H. Then for each t ≥ 0 we have Z t Z t ψ(X(t)) = ψ(X0) + hDψ(X(s−)),F (s)iH ds + hDψ(X(s−)), Ψ(s) dW (s)iH 0 0 Z Z + hDψ(X(s−)), f(s, ξ)iH dπb(s, ξ) (0,t] E REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5651

1 Z t + Tr[D2ψ(X(s−))Ψ(s)Ψ∗(s)] ds (26) 2 0 Z Z + (ψ(X(s−) + f(s, ξ)) − ψ(X(s−))) dπ(s, ξ) (0,t] E Z Z − hDψ(X(s−)), f(s, ξ)iH dπ(s, ξ) P-a.s. (0,t] E 2 We record below the special case in which ψ(u) := |u|H . Despite the fact that the second-order derivative of this map is not a Hilbert-Schmidt operator when H is infinite-dimensional, the terms on the right-hand side of (26) are still well-defined. By projecting X onto finite dimensional subspaces of H, applying Theorem 2.19 and 2 passing to the limit, one can show that equation (26) still holds for ψ(u) := |u|H . Corollary 1. Let X be as in (25). Then for all t ≥ 0 we have Z t Z t 2 2 |X(t)|H = |X0|H + 2 hX(s−),F (s)iH ds + 2 hX(s−), Ψ(s) dW (s)iH 0 0 Z Z Z t + 2 hX(s−), f(s, ξ)i dπ(s, ξ) + kΨ(s)k2 ds H b L2(U0,H) (0,t] E 0 Z Z 2 + |f(s, ξ)|H dπ(s, ξ) P-a.s. (27) (0,t] E More generally, one can show that (26) is also valid for the function ψ : H → R p defined by ψ(u) := |u|H , for every 2 ≤ p < ∞. See Chapter VI, Section 2 of [27] for further details and extensions of these results.

3. Existence for stochastic differential equations. The purpose of this section is to give a self-contained review of existence and uniqueness results for Hilbert space-valued SDEs with Lévy noise. In Subsection 3.1, assuming typical Lipschitz and linear growth conditions are satisfied globally, we consider solutions to SDEs that are defined globally, i.e., the solution exists for all time on a given interval [0,T ]. In Subsection 3.2, with local Lipschitz and linear growth conditions, we consider locally defined solutions to SDEs with Lévy noise, in which the solution exists on a random nontrivial subinterval of [0,T ] and may blow up with positive probability at a stopping time τ. We begin by introducing the setting for the SDE that we will consider in this section.

Let H be a real, separable Hilbert space with norm |·|H and inner product h·, ·iH . In this section we will consider the H-valued SDE ( dY = F (s, Y (s−)) ds + G(s, Y (s−)) dW (s) + R K(s, Y (s−), ξ) dπ(s, ξ), E0 b Y (0) = Y0, (28) with respect to a given stochastic basis (Ω, F, (Ft)t≥0, P, W, π), where W is an Ft- Wiener process, π is a Poisson random measure induced by a stationary Ft-Poisson point process and (Ω, F, (Ft)t≥0, P) satisfies the usual conditions (see Subsection 2.1). Remark 3. The noise in the SDE (28) is not the most general that can be considered. One can also add to the right-hand side of (28) the pure-jump noise term R R K(Y (s−), ξ) dπ(s, ξ), for some measurable function K: H × E → E. (0,t] E\E0 Once existence and uniqueness is established for equation (28), there is a standard 5652 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM method known as the piecing-out argument to obtain existence and uniqueness for (28) with the additional noise term R R K(Y (s−), ξ) dπ(s, ξ). We will post- (0,t] E\E0 pone discussion of this additional noise term and the piecing-out argument until Section 10. Returning to the setting of (28), we assume that W takes values in a real, separable Hilbert space U. We denote the covariance operator of W by Q and define 1/2 U0 := Q (U) as in Definition 2.13. We assume that π is a Poisson random measure on (0, ∞) × E for some measurable space (E, E). The intensity measure of π is assumed to be of the form dt⊗dν, where ν is a σ-finite measure on (E, E) (see Defi- nition 2.3). The set E0 is a fixed element of E, possibly with ν(E0) = ∞. As for the coefficients F , G and K in equation (28) we assume that F :Ω × [0, ∞) × H → H, G:Ω × [0, ∞) × H → L2(U0,H) and K :Ω × [0, ∞) × H × E → H are Borel measurable functions. Additional conditions will be imposed on F , G and K as part of criteria for establishing existence and uniqueness to equation (28), as well as showing that the stochastic integrals are well-defined. We assume that Y0 :Ω → H is an F0-measurable random variable. 2 Definition 3.1. Given Y0 ∈ L (Ω, F0, P; H), an H-valued, càdlàg, Ft-adapted process (Y (t))t≥0 is said to be a global pathwise solution to (28) on the interval [0,T ], if i) Y ∈ L2(Ω; L∞([0,T ]; H)) and, ii) for every t ∈ [0,T ], the following equation holds P-a.s.: Z t Z t Y (t) = Y0 + F (s, Y (s)) ds + G(s, Y (s−)) dW (s) 0 0 Z Z (29) + K(s, Y (s−), ξ) dπb(s, ξ). (0,t] E0 Let τ be a stopping time. The pair (Y, τ) is said to be a local pathwise solution to (28) up to stopping time τ if equation (29) is satisfied P-a.s. on the event {t < τ}. 3.1. Global existence for SDEs with global Lipschitz and linear growth condition. We review below a procedure for establishing existence and uniqueness of a solution to equation (28) in a series of results. First, in Theorem 3.3 we assume that F , G and K satisfy Lipschitz and linear growth conditions and 2 that E|Y0|H < ∞. In this setting a Picard iteration argument in the Hilbert space L2(Ω; L∞([0,T ]; H)) can be applied in order to establish existence for (28). We establish uniqueness of the solution in Proposition3. The linear growth and Lipschitz conditions on F , G and K are given below: we assume that there exists a constant C > 0 such that Z |F (s, x)|2 + kG(s, x)k2 + |K(s, x, ξ)|2 dν(ξ) ≤ C(1 + |x|2 ) H L2(U0,H) H H E0 (30) for all x ∈ H, s ≥ 0, ω ∈ Ω, and |F (s, x) − F (s, y)|2 + kG(s, x) − G(s, y)k2 H L2(U0,H) Z 2 2 + |K(s, x, ξ) − K(s, y, ξ)|H dν(ξ) ≤ C|x − y|H (31) E0 for all x, y ∈ H, s ≥ 0, ω ∈ Ω. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5653

We also assume that the following measurability conditions are satisﬁed for every x ∈ H: (ω, s) 7→ F (s, x) is a measurable mapping from Ω × [0, ∞) → H,

(ω, s) 7→ G(s, x) is a predictable mapping from Ω × [0, ∞) → L2(U0,H) and

(ω, s, ξ) 7→ K(s, x, ξ) is a P[0,T ] ⊗ E-measurable mapping from Ω × [0,T ] × E → H. (32) Lemma 3.2. Under conditions (30), (31) and (32) the stochastic integrals Z t Z t Z Z F (s, Y (s)) ds, G(s, Y (s−)) dW (s), and K(s, Y (s−), ξ) dπb(s, ξ) 0 0 (0,t] E0 (33) are well-deﬁned, Ft-adapted and have càdlàg versions for every Ft-adapted process Y ∈ L2(Ω × [0,T ], F ⊗ B([0,T ]), dP ⊗ dt; H) that has càdlàg sample paths P-a.s. Proof. Let Y ∈ L2(Ω × [0,T ], F ⊗ B([0,T ]), dP ⊗ dt; H) be adapted with càdlàg sample paths. Deﬁne Fe :Ω × [0,T ] → H, Ge :Ω × [0,T ] → L2(U0,H) and Ke :Ω × [0,T ] × E → H by ( G(s, Y (s−)) if s ∈ (0,T ] Fe(s) := F (s, Y (s)) Ge(s) := G(0,Y (0)) if s = 0, and ( K(s, Y (s−), ξ) if s ∈ (0,T ] Ke(s, ξ) := K(0,Y (0), ξ) if s = 0.

It is sufficient to show that Ge is predictable, Ke is P[0,T ] ⊗ E-measurable and Z T Z T E |F (s)|2 ds < ∞, E ||G(s)||2 ds < ∞, e H e L2(U0,H) 0 0 (34) Z T Z 2 and E |Ke(s, ξ)|H dν(ξ) ds < ∞. 0 E0 We observe that (34) follows from the growth condition (30) and the assump- R T 2 tion that E 0 |Y (s)| ds < ∞. To verify the measurability conditions we begin by noting that the process (Y (t−))t>0 is Ft-adapted and left continuous, and therefore predictable (see Lemma 2.7). The desired measurability properties can now be verified by approximating (Y (t−))t>0 with simple functions in the space 2 L (Ω × [0,T ], P[0,T ], dP ⊗ dt; H) (cf. Lemma 8.13 in [32]) and by using the Lip- schitz condition (31) and the measurability conditions (32). This shows that the three stochastic integrals in (33) are well-defined. The fact that they are adapted and have càdlàg versions follows from the construction of the stochastic integrals in Theorem 2.15 and Theorem 2.16. Theorem 3.3. In the setup above suppose that F , G and K satisfy conditions (30), 2 (31) and (32). Then for every Y0 ∈ L (Ω, F0, P; H) and every T > 0 there exists a global solution Y to equation (28) on the interval [0,T ] with initial condition Y0. The proof that we give below of Theorem 3.3 is mostly a combination of the arguments in Theorem 9.1 of [21] and Theorem 6.2.3 of [2]. One difference between our presentation and theirs is that we allow the coefficients F , G and K to be random and to depend on time. However, since we assume that the growth and Lipschitz 5654 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM conditions (30) and (31) hold uniformly over all (ω, s) ∈ Ω × [0, ∞), this extension does not change the main argument. Also, while the references [21] and [2] assume that the space H is finite-dimensional, we do not. Another difference between our setting and the setting in [21] and [2] is that we allow the Wiener process W to be infinite-dimensional. The technical tools that will be used in the proof of Theorem 3.3, for instance, the BDG inequality (inequality (18) and Proposition2), can be applied in the case where H and W are infinite-dimensional in the same way as in the finite-dimensional case. So allowing H and W to be infinite-dimensional in the setting does not introduce any additional difficulties to the proof of Theorem 3.3.

Proof of Theorem 3.3. We construct a solution using Picard iteration. Deﬁne Y0(t) ≡ Y0 and for each positive integer n deﬁne iteratively Z t Z t Yn(t) := Y0 + F (s, Yn−1(s)) ds + G(s, Yn−1(s−)) dW (s) 0 0 Z Z (35) + K(s, Yn−1(s−), ξ) dπb(s, ξ). (0,t] E0

By applying Lemma 3.2 inductively we see that each Yn is well-deﬁned, Ft-adapted, belongs to the space L2(Ω × [0,T ], F ⊗ B([0,T ]), dP ⊗ dt; H) and has càdlàg sam- R T 2 ple paths a.s. One can easily show that E 0 |Yn(s)|H ds < ∞ using the isometric formulas (15) and (21) and the growth assumption (30). We now pro- ∞ ceed to constructing a solution to (28) by showing that the sequence (Yn)n=1 2 ∞ is Cauchy in L (Ω; L ([0,T ]; H)). The fact that each Yn belongs to the space L2(Ω; L∞([0,T ]; H)) will follow from induction and the estimates below. For t ∈ [0,T ], we have Z t Yn+1(t) − Yn(t) = [F (s, Yn(s)) − F (s, Yn−1(s))] ds 0 Z t + [G(s, Yn(s−)) − G(s, Yn−1(s−))] dW (s) 0 Z Z + [K(s, Yn(s−), ξ) − K(s, Yn−1(s−), ξ)] dπb(s, ξ) (0,t] E0

=: I1(t) + I2(t) + I3(t). By the Cauchy-Schwarz inequality and (31) we have Z T 2 2 E sup |I1(t)|H ≤ T · E |F (s, Yn(s)) − F (s, Yn−1)(s)|H ds 0≤t≤T 0 Z T 2 . E|Yn(s) − Yn−1(s)|H ds, 0 where the hidden constant depends on T but not on n. In the inequality above, and throughout the article, we use . to mean inequality up to a hidden universal constant. By the Burkholder-Davis-Gundy inequality (inequality (18)) and the Lipschitz assumption (31), we have Z T E sup |I (t)|2 E kG(s, Y (s−)) − G(s, Y (s−))k2 ds 2 H . n n−1 L2(U0,H) 0≤t≤T 0 Z T 2 . E|Yn(s) − Yn−1(s)|H ds. 0 REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5655

Z T Z t1 Z tn−1 2 n 2 E sup |Yn+1(t)−Yn(t)|H ≤ c ··· E|Y1(tn)−Y0|H dtn ··· dt2 dt1. 0≤t≤T 0 0 0 (36) To estimate the innermost integrand we consider the n = 1 instance of (35), which reads Z s Z s Z Z Y1(s) − Y0 = F (r, Y0) dr + G(r, Y0) dW (r) + K(r, Y0, ξ) dπb(r, ξ). 0 0 (0,s] E0 Upon taking the H-norm squared on both sides, using the isometric formulas (13) and (21) and growth assumption (30) we ﬁnd that " # Z s 2 2 2 2 E|Y1(s) − Y0|H . s E sup |F (r, Y0)|H + E G(r, Y0) dW (r) r∈[0,s] 0 H Z Z 2 + E K(r, Y0, ξ) dπ(r, ξ) b H (0,s] E0 " # ≤ s2(1 + E|Y |2 ) + sE sup kG(r, Y )k2 0 H 0 L2(U0,H) r∈[0,s] Z s Z 2 + E |K(r, Y0, ξ)|H dν(ξ) dr 0 E0 2 . 1 + E|Y0|H . Applying this estimate in (36) and integrating yields n 2 n 2 T E sup |Yn+1(t) − Yn(t)|H . c (1 + E|Y0|H ) , (37) 0≤t≤T n! and hence (cT )n/2 kYn+1 − YnkL2(Ω;L∞([0,T ];H)) . √ , n! where the hidden constant does not depend on n. Note that the right-hand side of ∞ the estimate above is summable in n. As a result, the sequence (Yn)n=1 is Cauchy in the space L2(Ω; L∞([0,T ]; H)). Let Y be the limit of this sequence. We claim 5656 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM that Yn → Y uniformly on [0,T ] as n → ∞, P-a.s. Using Chebyshev’s inequality and (37) we see that 1 (4cT )n P sup |Yn+1(s) − Yn(s)|H > n . . 0≤s≤T 2 n! The right-hand side of the inequality above is summable in n, so the Borel-Cantelli lemma (see, e.g., Theorem 2.18 in [22]) implies that

h 1 i P sup |Yn+1(s) − Yn(s)|H > 2n infinitely often = 0. (38) 0≤s≤T Equation (38) can be stated in the following way: with probability one the inequality 1 sup |Yn+1(s) − Yn(s)|H ≤ n 0≤s≤T 2 ∞ holds for all but finitely many n. Using the triangle inequality we see that (Yn)n=1 converges in H uniformly on [0,T ], P-a.s. The limit is Y , of course. Since a uniform limit of càdlàg functions is also càdlàg we see that Y is càdlàg a.s. In addition, for each fixed t ∈ [0,T ] we have Yn(t) → Y (t) a.s., which implies that Y is Ft-adapted. 2 The convergence Yn(t) → Y (t) also holds in L (Ω; H). The final step is to pass to the limit in (35) and show that Y satisfies (29). Using the isometric formulas (13) and (21) and the Lipschitz assumption (31) it is easy to see that each term on the right-hand side of (35) converges in L2(Ω; H) to the corresponding term with Y in place of Yn−1. This establishes equation (29) for Y . Next is a uniqueness result for solutions to (28). Proposition 3. Assume that the hypotheses of Theorem 3.3 are satisfied. Then there exists a constant C = C(T ) > 0 such that for any two solutions Y and Z to 2 (28) with respective initial conditions Y0,Z0 ∈ L (Ω, F0, P; H) and for any event A ∈ F0 we have h 2 i 2 E 1A sup |Y (t) − Z(t)|H ≤ CE[1A|Y0 − Z0|H ]. (39) t∈[0,T ]

In particular, we have P[1{Y0=Z0}|Y (t) − Z(t)|H = 0 for all t ∈ [0,T ]] = 1, i.e., pathwise uniqueness holds for equation (28). Proof. Consider equation (29) for both Y and Z. Subtract the equation of Z from the equation of Y , then multiply by 1A and take the H-norm squared on both sides. Since A is F0-measurable, the indicator function 1A can be moved inside of the stochastic integrals. We easily obtain Z t 2 2 2 1A|Y (t) − Z(t)|H . 1A|Y0 − Z0|H + |F (s, Y (s)) − F (s, Z(s))|H 1A ds 0 Z t 2

+ [G(s, Y (s−)) − G(s, Z(s−))]1A dW (s) (40) 0 H Z Z 2 + [K(s, Y (s−), ξ) − K(s, Z(s−), ξ)]1A dπ(s, ξ) , b H (0,t] E0 P-a.s., for all t ∈ [0,T ]. Now ﬁx t0 ∈ [0,T ], take the supremum over t ∈ [0, t0] in the resulting inequality and take expectation to obtain

h 2 i 2 E 1A sup |Y (t) − Z(t)|H . E[1A|Y0 − Z0|H ] t∈[0,t0] REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5657

Z t0 2 + E|F (s, Y (s)) − F (s, Z(s))|H 1A] ds 0 h Z t 2 i

+ E sup [G(s, Y (s−)) − G(s, Z(s−))]1A dW (s) t∈[0,t0] 0 H h Z Z 2 i + E sup [K(s, Y (s−), ξ) − K(s, Z(s−), ξ)]1A dπ(s, ξ) 0 b H t∈[0,t ] (0,t] E0 2 := E[1A|Y0 − Z0|H ] + J1 + J2 + J3. (41) Using the Lipschitz condition (31) we see that Z t0 Z t0 2 2 J1 . E[|Y (s) − Z(s)|H 1A] ds ≤ E[1A sup |Y (t) − Z(t)|H ] ds. 0 0 t∈[0,s]

For J2 we use the BDG inequality (inequality (18)) and Lipschitz assumption (31) to obtain Z t0 J E kG(Y (s)) − G(Z(s))k2 1 ds 2 . L2(U0,H) A 0 Z t0 2 . E |Y (s) − Z(s)|H 1A ds 0 Z t0 2 ≤ E[1A sup |Y (t) − Z(t)|H ] ds. 0 t∈[0,s]

For J3 we use the BDG inequality (Proposition2) and (31) to obtain Z Z 2 J3 . E |K(s, Y (s−), ξ) − K(s, Z(s−), ξ)|H 1A dπ(s, ξ) 0 (0,t ] E0 Z t0 Z 2 = E |K(s, Y (s−), ξ) − K(s, Z(s−), ξ)|H 1A dν(ξ) ds 0 E0 Z t0 2 . E |Y (s) − Z(s)|H 1A ds 0 Z t0 2 ≤ E[1A sup |Y (t) − Z(t)|H ] ds. 0 t∈[0,s] Returning to (41) we find that Z t0 2 2 2 E[1A sup |Y (t) − Z(t)|H ] . E[1A|Y0 − Z0|H ] + E[1A sup |Y (t) − Z(t)|H ] ds. t∈[0,t0] 0 t∈[0,s] (42) Since the hidden constant does not depend on Y , Z, t0 or A we deduce (39) using the deterministic Gronwall inequality. Remark 4. Proposition3 shows that there exists a well-defined solution map for 2 equation (28) from the space L (Ω, F0, P; H) of initial conditions to the space L2(Ω; L∞([0,T ]; H)). By taking A := Ω in (39) we see that the solution map is Lipschitz continuous as a function of the initial data. It is also possible to establish existence and uniqueness of solutions to equation (28), as well as continuity in initial data, by making a fixed-point argument as in Theorem 9.29 of [32]. 5658 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

We now state a related local uniqueness result that will be useful when consid- ering local solutions to equation (28). Proposition 4. Assume that the hypotheses of Theorem 3.3 are satisﬁed. Let (Y, τ) and (Z, τ) be two local solutions to (28) up to the same Ft-stopping time 2 τ with respective initial conditions Y0,Z0 ∈ L (Ω, F0, P; H). Then there exists a constant C = C(T ) > 0 such that for every event A ∈ F0, we have

h 2 i 2 E 1A sup |Y (t) − Z(t)|H ≤ CE[1A|Y0 − Z0|H ]. (43) t∈[0,T ∧τ) In particular, we have P 1{Y0=Z0}|Y (t) − Z(t)|H = 0 for all t ∈ [0, τ) = 1. Proof. The proof of Proposition4 is a slight modiﬁcation of the proof of Proposition 3, so we will not rewrite all of the details. The main diﬀerence is that instead of taking the supremum over t ∈ [0, t0] prior to (41), one takes the supremum over t ∈ [0, t0 ∧ τ). The BDG inequality and the Lipschitz condition (31) can be applied exactly as in the proof of Proposition (3). This leads to the following analogue of (42):

p The next result shows that if the initial data Y0 belongs to L (Ω, F0, P; H) for some p ∈ [2, ∞), and if the function K satisﬁes a certain growth condition depending on p, then the solution Y to (28) belongs to the space Lp(Ω; L∞([0,T ]; H)). Proposition 5. Assume that the hypotheses of Theorem 3.3 are satisﬁed. Suppose p that E|Y0|H < ∞ for some p ≥ 2 and assume that there exists a constant C > 0 such that Z p p |K(s, x, ξ)|H dν(ξ) ≤ C(1 + |x|H ) for all x ∈ H, s ≥ 0. (45) E0 Then the solution Y to equation (28) belongs to the space Lp(Ω; L∞([0,T ]; H)). Furthermore, there exists a constant c > 0 depending only on T , p and the constants in (30) and (45), such that

p p E sup |Y (s)|H ≤ c(1 + E|Y0|H ). (46) s∈[0,T ] Proof. The p = 2 case is contained in the statement and proof of Theorem 3.3, so we take p > 2 from now on. We apply the Itô formula (Theorem 2.19) to the p solution Y of equation (28) using the function ψ : H → R deﬁned by ψ(x) := |x|H up to the stopping time

τm := inf{t > 0 : |Y (t)|H > m} ∧ T, for a ﬁxed positive integer m. The derivatives Dψ : H → H∗ and D2ψ : H → B(H) p−2 2 p−2 of ψ are given by (Dψ)(x) = p|x|H hx, ·iH and (D ψ)(x) = p|x|H I + p(p − p−4 2)|x|H hx, ·iH x. The Cauchy-Schwarz inequality shows that the second term in REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5659 the formula for (D2ψ)(x) is continuous at x = 0 even when 2 < p < 4. Applying the Itô formula yields

Z t∧τm p p p−2 |Y (t ∧ τm)|H = |Y0|H + p|Y (s)|H hY (s),F (s, Y (s))iH ds 0 Z t∧τm p−2 + p|Y (s)|H hY (s),G(s, Y (s−)) dW (s)iH 0 p Z t∧τm + |Y (s)|p−2 kG(s, Y (s−))k2 ds H L2(U0,H) 2 0 p(p − 2) Z t∧τm + |Y (s)|p−4|G(s, Y (s−))∗Y (s)|2 ds H U0 2 0 Z Z p−2 + p|Y (s−)|H hY (s−),K(s, Y (s−), ξ)iH dπb(s, ξ) (0,t∧τm] E0 Z Z h p p + |Y (s−) + K(s, Y (s−), ξ)|H − |Y (s−)|H (0,t∧τm] E0 p−2 i − p|Y (s−)|H hY (s−),K(s, Y (s−), ξ)iH dπ(s, ξ) p = |Y0|H + I1(t) + I2(t) + I3(t) + I4(t) + I5(t), (47) P-a.s. for all t ∈ [0,T ],

∗ where G(s, Y (s−)) : H → U0 denotes the adjoint G(s, Y (s−)). We estimate I1 using the Cauchy-Schwarz inequality and the growth condition (30) and obtain

Z t∧τm h i p−1 E sup |I1(s)|H ≤ pE |Y (s)|H |F (s, Y (s))|H ds s∈[0,t] 0 (48) Z t∧τm p−1 ≤ CpE |Y (s)|H (1 + |Y (s)|H ) ds. 0 From inequality (48) we obtain

Z t ! h i p E sup |I1(s)|H . 1 + E sup |Y (r)|H ds (49) s∈[0,t] 0 r∈[0,s∧τm] and the hidden constant only depends on p and the constant in (30). For I2 we use the BDG inequality (18) to obtain

h i Z t∧τm 1/2 E sup |I (s)| E |Y (s)|2(p−2)|G(s, Y (s−))∗Y (s)|2 ds . (50) 2 H . H U0 s∈[0,t] 0

We also use the fact that the L2(U0, R)-norm of a linear functional on U0 is just the U0-norm of the corresponding vector by the Riesz representation theorem. Since the Hilbert-Schmidt norm dominates the operator norm we have ∗ 2 ∗ 2 2 |G(s, Y (s−)) Y (s)|U ≤ ||G(s, Y (s−)) ||L (H,U )|Y (s)|H 0 2 0 (51) = kG(s, Y (s−))k2 |Y (s)|2 . L2(U0,H) H Applying (51) in (50) we ﬁnd that h i E sup |I2(s)|H . s∈[0,t] 5660 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

1/2 Z t∧τm E sup |Y (s)|p/2 |Y (s)|p−2 kG(s, Y (s−))k2 ds . H H L2(U0,H) s∈[0,t∧τm] 0 Using Cauchy-Schwarz inequality we ﬁnd that there exists a positive constant C > 0 such that

h i 1 p E sup |I2(s)|H ≤ E sup |Y (s)|H s∈[0,t] 4 s∈[0,t∧τm] Z t∧τm + CE |Y (s)|p−2 kG(s, Y (s−))k2 ds. H L2(U0,H) 0 Using the growth condition (30) on the right-hand side of the above inequality we ﬁnd that Z t ! h i 1 p p E sup |I2(s)|H ≤ E sup |Y (s)|H +C +C E sup |Y (r)|H ds, s∈[0,t] 4 s∈[0,t∧τm] 0 r∈[0,s∧τm] (52) for a possibly larger constant C (which depends on T ). In I3 we can use inequality (51) to estimate the second term. The resulting bound has the same form as the ﬁrst term in I3. Using (30) we can then conclude that Z t ! h i p E sup |I3(s)|H . 1 + E sup |Y (r)|H ds, (53) s∈[0,t] 0 r∈[0,s∧τm] where again the hidden constant only depends on p, T and the constant in (30). We apply the BDG inequality (Proposition2) to I4 and obtain h i E sup |I4(s)|H s∈[0,t] Z Z 1/2 p−2 2 . E (p|Y (s−)|H hY (s−),K(s, Y (s−), ξ)i) dπ(s, ξ) (0,t∧τm] E0 Z Z 1/2 2p−2 2 . E |Y (s−)|H |K(s, Y (s−), ξ)|H dπ(s, ξ) (0,t∧τm] E0 Z Z !1/2 p/2 p−2 2 ≤ E sup |Y (s)|H |Y (s−)|H |K(s, Y (s−), ξ)|H dπ(s, ξ) . s∈[0,t∧τm] (0,t∧τm] E0 Using Young’s inequality we see that there exists a positive constant C such that ! h i 1 p E sup |I4(s)|H ≤ E sup |Y (s)|H s∈[0,t] 4 s∈[0,t∧τm] Z Z p−2 2 + CE |Y (s−)|H |K(s, Y (s−), ξ)|H dπ(s, ξ). (0,t∧τm] E0 In the second term on the right-hand side of the inequality above, the random measure dπ can be replaced in the expectation by its intensity measure dν ⊗ ds by Theorem 2.16. Then an application of inequality (30) yields ! h i 1 p E sup |I4(s)|H ≤ E sup |Y (s)|H s∈[0,t] 4 s∈[0,t∧τm]

Z t∧τm p−2 2 + CE |Y (s−)|H (1 + |Y (s−)|H ) ds. 0 REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5661 ! 1 p ≤ E sup |Y (s)|H + C 4 s∈[0,t∧τm] Z t ! p + C E sup |Y (r)|H ds. (54) 0 r∈[0,s∧τm]

In the estimate of I5 we make use of the inequality

p p p−2 p−2 2 p |y + k|H − |y|H − p|y|H hy, kiH . (|y|H |k|H + |k|H ) for all y, k ∈ H, (55) which is easily obtained from the fundamental theorem of calculus (see, e.g., equation (4.9) in [6]). The hidden constant in inequality (55) depends only on p. Using inequality (55) we see that h i Z Z p p E sup |I5(s)|H ≤ E |Y (s−) + K(s, Y (s−), ξ)|H − |Y (s−)|H s∈[0,t∧τm] (0,t] E0

p−2 − p|Y (s−)|H hY (s−),K(s, Y (s−), ξ)iH dπ(s, ξ) Z Z h p−2 2 p i . E |Y (s−)|H |K(s, Y (s−), ξ)|H + |K(s, Y (s−), ξ)|H dπ(s, ξ) (0,t∧τm] E0 Z Z h p−2 2 p i = E |Y (s−)|H |K(s, Y (s−), ξ)|H + |K(s, Y (s−), ξ)|H dν(ξ) ds. (0,t∧τm] E0 (56) Using (30), (45) and Young’s inequality we obtain

Z t∧τm h i p E sup |I5(s)|H . 1 + E |Y (s−)|H ds. (57) s∈[0,t] 0 p p Next we replace |Y (s−)|H with supr∈[0,s∧τm] |Y (r)|H on the right-hand side of the inequality above and integrate to time t. Using this estimate, along with (49), (52), (53) and (54), in (47) we find that there exists a constant C > 0 such that Z t ! h p i p p E sup |Y (s)|H ≤ E|Y0|H + C + C E sup |Y (r)|H ds. s∈[0,t∧τm] 0 r∈[0,s∧τm] Using Gronwall’s inequality we deduce that p p E sup |Y (s)|H ≤ c(1 + E|Y0|H ) s∈[0,T ∧τm] for a constant c that depends on T and p but not on m. Since τm ↑ T as m → ∞ we deduce (46) and conclude that Y ∈ Lp(Ω; L∞([0,T ]; H)). 3.2. Local existence for SDE with local Lipschitz and linear growth conditions. In this section we continue to consider the SDE (28), which we now restate for convenience: ( dY = F (s, Y (s)) ds + G(s, Y (s−)) dW + R K(s, Y (s−), ξ) dπ(s, ξ) E0 b (58) Y (0) = Y0. In contrast to Subsection 3.1, we weaken the growth and Lipschitz conditions (30) and (31) on the coefficients F,G and K to versions that are only assumed to hold locally in H and we also remove integrability conditions from the initial condition Y0. Instead of global existence for (58) on a given time interval [0,T ], we only 5662 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM obtain local existence up to an Ft-stopping time τ, at which point the H-norm of the solution could explode. We give the local versions of conditions (30) and (31) and then prepare the setting for proving local existence to (58). We impose the same conditions on the stochastic basis (Ω, F, (Ft)t≥0, P, W, π) as in Subsection 3.1. It will be important in this section to recall that π is assumed to be the Poisson random measure induced by a stationary Ft-Poisson point process Ξ on a measurable space (E, E) as in (5), i.e., π has the form π = P δ . For the coefficients F , G and K in (58), s∈DΞ (s,Ξ(s)) we now assume that for each positive integer r there exists a constant Cr > 0 such that Z |F (s, x)|2 + kG(s, x)k2 + |K(s, x, ξ)|2 dν(ξ) ≤ C (1 + |x|2 ), (59) H L2(U0,H) H r H E0 for all (ω, s) ∈ Ω × [0, ∞) and all x ∈ H with |x|H ≤ r and |F (s, x) − F (s, y)|2 + kG(s, x) − G(s, y)k2 H L2(U0,H) Z (60) 2 2 + |K(s, x, ξ) − K(s, y, ξ)|H dν(ξ) ≤ Cr|x − y|H , E0 for all (ω, s) ∈ Ω × [0, ∞) and all x, y ∈ H with |x|H , |y|H ≤ r. We continue to assume that F , G and K satisfy the measurability condition (32). We assume that Y0 :Ω → H is an F0-measurable random variable. For each fixed positive integer r we define a cutoff function θr : H → H by r θr(x) := x if |x|H ≤ r and θr(x) := x otherwise. Define Fr :Ω × [0, ∞) × H → |x|H H, Gr :Ω × [0, ∞) × H → L2(U0,H) and Kr :Ω × [0, ∞) × H × E0 → H by Fr(s, x) := F (s, θr(x)), Gr(s, θr(x)) := G(s, θr(x)) and Kr(s, x, ξ) := K(s, θr(x), ξ) for all s ≥ 0, x ∈ H and ξ ∈ E0. Since θr is Lipschitz and |θr(x)|H ≤ |x|H for all x ∈ H, it is easy to see from (59) and (60) that the modified coefficients Fr, Gr and Kr satisfy the growth and Lipschitz conditions (30) and (31) globally. By Theorem 3.3 there exists a unique adapted, càdlàg H-valued process Yr such that the equation Z t Z t

Yr(t) = Y01{|Y0|H

τr := inf{t > 0 : |Yr(t)|H > r} ∧ T (62) is an Ft-stopping time (see, e.g., Proposition 4.6 in Chapter I of [35]). For each t ∈ [0,T ] we have Z t Z t

Yr(t) = Y01{|Y0|H

Yr1 (t) = Yr2 (t) for all t ∈ [0, τr1 ∧ τr2 ), P-a.s. on the event {|Y0|H < r1}. (64) ∞ Our goal is to use the local solutions ((Yr, τr))r=1 to (58) with initial conditions ∞ Y 1 to construct a local solution Y to (58) with initial condition Y . 0 {|Y0|H r}. To avoid this kind of triviality we will only look at properties of τr on events of the form {|Y0|H < R} for r > R. Below we show that on each event {|Y0|H < R}, the stopping times ∞ (τr)r=R+1 are positive a.s. and increase monotonically. Lemma 3.4. Suppose that F , G and K satisfy conditions (32), (59) and (60), let Y0 :Ω → H be F0-measurable and deﬁne Yr and τr as above. Then for every positive integer R we have S∞ i) P [ r=R{τr = 0, |Y0|H < R}] = 0, ∞ ii) the sequence (τr)r=R increases monotonically P-a.s. on the event {|Y0|H < R}, and iii) Yr1 (t) = Yr2 (t) for all t ∈ [0, τr1 ], P-a.s. for all r2 ≥ r1 ≥ R on the event {|Y0|H < R}.

Proof.i ) This follows immediately from the fact that Yr is right-continuous at time t = 0, P-a.s.

ii) Let r2 > r1 ≥ R be positive integers. By (64) we have Yr1 = Yr2 on the

interval [0, τr1 ∧ τr2 ), P-a.s. on the event {|Y0|H < R}. We claim that equality

also holds at the time τr1 ∧ τr2 . Note that both Yr1 and Yr2 are continuous at

τr1 ∧τr2 if and only if τr1 ∧τr2 does not belong to the domain DΞ of the Poisson

point process Ξ. In this case, we clearly have Yr1 (τr1 ∧ τr2 ) = Yr2 (τr1 ∧ τr2 ).

On the other hand, if τr1 ∧ τr2 ∈ DΞ, then both Yr1 and Yr2 have a jump at

time τr1 ∧ τr2 of size ∆Yrj (τr1 ∧ τr2 ) = Krj τr1 ∧ τr2 ,Yrj (τr1 ∧ τr2 −), Ξ(τr1 ∧ τr2 ) , for j = 1, 2. Since

r1 ≥ |Yr1 (t)|H = |Yr2 (t)|H

for all t ∈ [0, τr1 ∧ τr2 ), we see that

∆Yrj (τr1 ∧ τr2 ) = K(τr1 ∧ τr2 ,Yr1 (τr1 ∧ τr2 −), Ξ(τr1 ∧ τr2 )), for j = 1, 2. In particular, the right-hand side of the equation above does not depend on j. This shows that

Yr1 (τr1 ∧ τr2 ) = Yr2 (τr1 ∧ τr2 ) P-a.s. on {|Y0|H < R}. (65)

Now, by right-continuity of Yrj we have

|Yrj (τrj )|H ≥ rj, for j = 1, 2.

If τr2 < τr1 , then we would have

|Yr1 (τr2 )|H = |Yr2 (τr2 )|H ≥ r2 > r1,

which is absurd. This shows that for all r2 > r1 ≥ R we have τr1 ≤ τr2 , P-a.s. on the event {|Y0|H < R}. By taking a countable intersection of events ∞ with full measure we see that the sequence (τr)r=R increases monotonically P-a.s. on the event {|Y0|H < R}. 5664 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

iii) Let r2 ≥ r1 ≥ R be positive integers. By (64) we have Yr1 = Yr2 on the interval

[0, τr1 ∧ τr2 ), P-a.s. on the event {|Y0|H < R}. By part ii) and equation (65)

we have Yr1 (τr1 ) = Yr2 (τr1 ), P-a.s. on the event {|Y0|H < R}. Using the results in Lemma 3.4 we can now define a local pathwise solution to equation (58) for any F0-measurable initial condition Y0 :Ω → H. That lemma shows that on the event {|Y0|H < R}, for every r2 ≥ r1 ≥ R, the graph of Yr1 as a function of t ∈ [0, τr1 ] is extended by the graph of Yr2 as a function of t ∈ [0, τr2 ]. We therefore define the local solution to equation (58) with initial condition Y0 as the maximal extension of these graphs. We now give precise details of the construction. We define the Ft-stopping time

τ := lim sup τr. (66) r→∞ Lemma 3.4 shows that

τ = sup τr, P-a.s., (67) r>|Y0|H so [0, τ) = S [0, τ ] a.s. We do not use (67) as the deﬁnition of τ because it r>|Y0|H r is not clear a priori that the right-hand side of (67) is an Ft-stopping time. Given ∞ t ∈ [0,T ) with t < τ, the sequence (Yr(t))r=1 stabilizes in H when r > |Y0|H , P-a.s., by Lemma 3.4. We deﬁne an H-valued stochastic process (Y (t))t∈[0,T ) by ( lim Yr(t) if t < τ Y (t) := r→∞ (68) 0 if t ≥ τ.

It is clear that Y is adapted to the filtration (Ft)t≥0 and that Y is càdlàg on [0, τ) almost surely. Since Y (t) = Yr(t) for all t ∈ [0, τr] a.s. on the event {|Y0|H < r}, we see that Y is a local solution to equation (58) with initial condition Y0 up to time τ. In order to justify the definition of Y in (68), we must show that τ is positive almost surely. We do this next and also show that the H-norm of Y blows up at time τ on the event {τ < T }. Proposition 6. Suppose that F , G and K satisfy conditions (32), (59) and (60). Let Y0 :Ω → H be F0-measurable and define Yr and τr as above. Let τ be as in (67) and let Y be as in (68). Then the following statements hold: i) P[τ = 0] = 0 . S∞ ii) P [ r=1{τr = τ} ∩ {|Y0|H < r} ∩ {τ < T }] = 0. iii) sup |Y (t)|H = ∞, P-a.s. on the event {τ < T }. t∈[0,τ) h 2 i iv) If E supt∈[0,τ) |Y (t)|H < ∞, then P[τ = T ] = 1. S∞ Proof.i ) Since P [ R=1{|Y0|H < R}] = 1 it suffices to show that P[τ = 0, |Y0|H < R] = 0 for each positive integer R. On the event {|Y0|H < R} we have τ = supr≥R τr a.s. by (67). So,

P[τ = 0, |Y0|H < R] ≤ P[τR = 0, |Y0|H < R] and the right-hand side of the inequality above is zero by part i) of Lemma 3.4. ii) We need to show that P[τR = τ, |Y0|H < R, τ < T ] = 0 for each ﬁxed positive integer R. For r ≥ R we have τr ≥ τR a.s. on the event {|Y0|H < R} by part REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5665

ii) of Lemma 3.4, whence ∞ h \ i P[τR = τ, |Y0|H < R, τ < T ] = P {τr = τR, |Y0|H < R, τr < T } . (69) r=R

Consider the random positive integer R0 := 1 + d|YR(τR)|H e, where for x ∈ R, dxe denotes the smallest integer that is no less than x. Almost surely on the event {|Y0|H < R} we have

sup |Yr(t)|H = sup |YR(t)|H < R0 t∈[0,τR] t∈[0,τR]

for all r > R. Since each |Yr|H is right-continuous a.s., it follows that τR < τR0 a.s. on the event {|Y0|H < R, τR < T }. This shows that the probabilities in (69) are zero. iii) Part ii) shows that a.s. on the event {τ < T } we have either τR < τ or |Y0|H ≥ R for every positive integer R. In particular, we have τR < τ all but ﬁnitely often a.s. on {τ < T }. So, P-a.s. on the event {τ < T } we have that |Y (τR)|H ≥ R for every positive integer R, which proves iii). iv) This follows immediately from part iii).

The results above lead to the following local existence result for equation (58): Theorem 3.5. Suppose that H is a separable Hilbert space, and that F , G and K satisfy conditions (32), (59) and (60) and let Y0 :Ω → H be F0-measurable. Then there exists an Ft-adapted H-valued process (Y (t))t∈[0,T ] and an Ft-stopping time τ such that τ > 0 a.s., Y is càdlàg in H on [0, τ) P-a.s., and the pair (Y, τ) is a local solution to equation (58), i.e., Z t Z t Y (t) = Y0 + F (s, Y (s)) ds + G(s, Y (s−)) dW (s) 0 0 Z Z (70) + K(s, Y (s−), ξ) dπb(s, ξ), (0,t] E0 for all t ∈ [0, τ) a.s., and

sup |Y (t)|H = ∞ P-a.s. on {τ < T }. t∈[0,τ) Furthermore, the pair (Y, τ) is pathwise unique in the sense that it extends every local solution to equation (58).

Part 2. An abstract evolution equation with Lévy Noise 4. Setting and main result. In this section we deﬁne the setting for the SPDE that will be the main focus of the rest of the article. We consider an SPDE with Lévy noise in an abstract form that is typical in classical or geophysical ﬂuid mechanics. The main SPDE, stated in the Introduction as equation (1), is restated below for convenience.  du + [Au + B(u, u) + F (u)] dt = G(u(t−)) dW (t)   + R K(u(t−), ξ) dπ(t, ξ) E0 b (71) + R K(u(t−), ξ) dπ(t, ξ),  E\E0  u(0) = u0. 5666 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Our goal is to establish local existence and uniqueness of a pathwise solution for equation (1). We begin by setting up the precise mathematical framework in which equation (1) and each of its terms is well-deﬁned in Subsection 4.1. The main result is stated in Subsection 4.2 as Theorem 4.3 after the precise notion of local pathwise solution to (1) is deﬁned. In Subsection 4.3 we outline the plan for establishing the main result.

4.1. Functional framework. We consider separable, real Hilbert spaces V ⊂ H, and assume that the embedding is dense and compact. We may thus define the Gelfand inclusions V ⊂ H ⊂ V 0, where V 0 is the dual of V , and H has been identified with its dual H0. We denote by (·, ·), | · |, ((·, ·)), and k · k the norms and inner products of H, and V respectively. The duality pairing between V 0 and V will 0 0 be denoted by h·, ·i. The inclusion H ⊂ V sends a vector u ∈ H to (u, ·)|V ∈ V . For example, V may be a closed subspace of H1(O) for some open set O ⊆ Rn, and H may be the closure of V in L2(O). We now give the precise assumptions on each of the terms appearing in (1), beginning with the linear term A. We assume that A : D(A) ⊂ V → H is an unbounded, densely defined, bijective, linear operator such that (Au, v) = ((u, v)) for all u, v ∈ D(A). Under these conditions on A, the space D(A) is complete under the Hilbertian norm |u|D(A) := |Au|. In the example above, where H is a subspace of L2(O) and V is a subspace of H1(O), the role of A can be played by the negative Laplacian associated with suitable boundary conditions. It is clear that for suitable boundary conditions, the operator A is symmetric. The operator A can be viewed as a linear continuous operator from V to V 0 via hAu, vi := ((u, v)), for all u, v ∈ V. (72) It is immediate from this equation that A: V → V 0 is bounded with norm 1. Furthermore, it is easy to show that the linear operator A−1 : H → V is bounded using the closed graph theorem. Indeed, suppose that Aun → Au in H (recall that A is assumed to be surjective, so any convergent sequence in H can be written this −1 way) and suppose that A (Aun) = un → v in V . We need to show that u = v. For any w ∈ V we have

((un − u, w)) = hAun − Au, wi → 0, which means that un → u weakly in V . Since we assume un → v strongly in V it follows that u = v, so A−1 : H → V is continuous by the closed graph theorem. Since we assume that V is compactly embedded in H, it follows that A−1 is a compact operator on H. Indeed, A−1 maps bounded sequences in H to bounded sequences in V , which have H-convergent subsequences by the compact embedding of V in H. Since A is assumed to be symmetric and surjective, we easily see that A−1 is self-adjoint. From the relation (A−1(Au), Au) = (Au, u) = kuk2 for all u ∈ D(A) we see that A−1 is a positive operator. By the spectral theorem for positive compact ∞ operators there exists an orthonormal basis (Awk)k=1 of H consisting of eigenvectors of A−1 whose corresponding eigenvalues are positive and tend to 0 as k → ∞. The ∞ sequence (wk)k=1 lies in D(A) and forms an orthogonal basis of H. Furthermore, each wk is an eigenvector of A whose corresponding eigenvalue λk is positive and λk → ∞ as k → ∞. Note that ∞ ∞ 1/2 n X X 2 o D(A ) := αkwk ∈ H : αkλk < ∞ k=1 k=1 REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5667 is nothing but the space V . Under the conditions given above, one can show that the embedding D(A) ⊂ D(A1/2) = V is compact (see e.g. [23]). We assume that B : V × D(A) → V 0 is bilinear and continuous. Furthermore, we assume that B maps the subspace D(A) × D(A) continuously to H. We assume that B possesses the cancellation property hB(v, u), ui = 0, for all v ∈ V and u ∈ D(A). (73)

We also assume that there exist constants C, c0 > 0 such that |hB(v, u), wi| ≤ Ckvk · |Au| · kwk, (74) for all v, w ∈ V and u ∈ D(A) and 1/2 1/2 1/2 1/2 |hB(v, u), wi| ≤ c0kvk |Av| kuk |Au| |w|, (75) for all v, u ∈ D(A) and w ∈ H. We will write B(u) := B(u, u) for u ∈ D(A). In the example above, where V is a subspace of H1(O) and H is a subspace of L2(O), we have in mind for B the function B(v, u) := (v · ∇)u, corresponding to the nonlinear terms in the Navier- Stokes equation. The details of the application of the general results to the primitive equations (see e.g. [33] for the deterministic setting) will be given elsewhere. We assume that the nonlinear operator F in equation (1) satisﬁes F : V → H. (76) The following local growth and local Lipschitz assumptions are imposed on F : for every R > 0 there exists a constant CR > 0 such that

|F (u)| ≤ CR(1 + kuk), for all u ∈ V with kuk ≤ R, (77a) and

|F (u) − F (v)| ≤ CRku − vk, for all u, v ∈ V with max(kuk, kvk) ≤ R. (77b) Before discussing the coefficients G and K of the noise terms in equation (1) we discuss the assumptions on the noise. The noise processes will be defined on a fixed probability space (Ω, F, P). In equation (1) we assume that W is a Wiener process taking values in a real, separable Hilbert space U. We denote by Q the 1/2 covariance operator of W (see Definition 2.10) and we set U0 := Q (U) as in Definition 2.13. In equation (1) we assume that π is a Poisson random measure on (0, ∞) × E arising from a stationary Poisson point process Ξ on a measurable space (E, E) (see Definition 2.3). The intensity measure of π has the form dν ⊗ dt for some σ-finite measure ν on (E, E). In equation (1) we fix a set E0 ∈ E such that ν(E \ E0) < ∞. Furthermore, we assume that W is independent of π. Under this condition it is possible to construct a filtration (Ft)t≥0 on (Ω, F, P) such that W is an Ft-Wiener process (see Definition 2.8), Ξ is an Ft-Poisson point process and such that the filtered probability space (Ω, F, (Ft)t≥0, P) satisfies the usual conditions (as explained in [8]). We assume that the coefficients G and K in equation (1) satisfy the following conditions:

• G: H → L2(U0,H) and G maps V into L2(U0,V ), and • K : H × E → H is a Borel measurable function and K maps V × E into V . 5668 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

We now impose growth and Lipschitz conditions on G and K. We assume that there exists a constant C > 0 such that Z kG(v)k2 + kK(v, ξ)k2 dν(ξ) ≤ C(1 + kvk2) for all v ∈ V, (78) L2(U0,V ) E0 Z kG(u) − G(v)k2 + kK(u, ξ)−K(v, ξ)k2 dν(ξ) ≤ Cku−vk2 for all u, v ∈ V, L2(U0,V ) E0 (79) and Z kK(v, ξ)k4 dν(ξ) ≤ C(1 + kvk4), for all v ∈ V. (80) E0 We also assume that both (78) and (79) hold with the space V replaced by H. Note that measurability condition (32) is satisfied by G and K because each is deterministic and does not depend on time. So, Lemma 3.2 implies that the stochastic integrals R t G(u(s−)) dW (s) and R R K(u(s−), ξ) dπ(s, ξ) appearing on the right-hand 0 (0,t] E0 b side of equation (1) are well-defined for every adapted, càdlàg, H-valued process u R T 2 such that E 0 |u(s)| ds < ∞. The stochastic integral R t R K(u(s−), ξ) dπ(s, ξ) appearing as the final term 0 E\E0 on the right-hand side of equation (1) is to be understood in the sense of part (i) of Theorem 2.16. We assume that K: H × E → V is a measurable function. In contrast to K, there will be no growth or Lipschitz assumptions on K. We introduce the additional term with coefficient K on the right-hand side of (1) to represent the influence of large jumps from Lévy noise. If the Lévy noise is not square integrable, then the influence of large jumps cannot be represented without this additional term; see e.g., [8] or [32]. For an example of K, consider a U-valued Lévy process L with Lévy measure ν, let E := U \{0} and E0 := {ξ ∈ U : 0 < |ξ|U < 1} and define K: H × E → H by

K(u, ξ) := |ξ|U · u, for all u ∈ H.

We remark that the assumption ν(E \E0) < ∞ is satisﬁed in this case; see Theorem 4.23 in [32] for a proof of this fact.

4.2. Statement of the main result. Having now deﬁned each of the terms that appear in equation (1), we now give the precise deﬁnition of the kind of solution that will be considered and then state the main result.

Definition 4.1. A stochastic basis is a tuple S := (Ω, F, (Ft)t≥0, P, W, π), where (Ω, F, (Ft)t≥0, P) is a filtered probability space satisfying the usual conditions, W is a U-valued Wiener process and π is a Poisson random measure satisfying the same conditions listed above in Sections2 and3. The stochastic basis contains all of the information about the probability space and the noise. As described in Sections2 and3 we impose typical assumptions on the noise processes W and π and on the filtration (Ft)t≥0 to ensure the stochastic integrals appearing on the right-hand side of equation (1) exist. Given a stochastic basis, our goal is to find a solution to equation (1) that lives on the same probability space. This type of a solution, referred to as pathwise, is defined next. A weaker type of solution in law, known as a martingale solution, will also be considered in this article and is defined precisely in Definition 4.4. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5669

Definition 4.2. Fix a stochastic basis (Ω, F, (Ft)t≥0, P, W, π) and a random variable u0 :Ω → V that is F0-measurable. We say that a pair (u, τ), where τ is an Ft-stopping time is a local pathwise solution to (1) if u is a D(A)-valued Ft-adapted process and the sample paths of u are càdlàg in H, and if P-a.s. we have Z t u(t) + [Au(s) + B(u(s))+ F (u(s))] ds 0 Z t = u0 + G(u(s−)) dW (s) 0 Z Z (81) + K(u(s−), ξ) dπb(s, ξ) (0,t] E0 Z Z + K(u(s−), ξ) dπ(s, ξ) (0,t] E\E0 in H for all t ∈ [0, τ). We say that a local pathwise solution (u, τ) is maximal if sup |u(t)| = ∞ P-a.s. on the event {τ < T }. (82) t∈[0,τ) Equation (81) is the correct mathematical formulation of the shorthand equation (1). We are now prepared to state the main result. Theorem 4.3. Let A, B, F, G, K, K satisfy the assumptions laid out in Section 4.1. Let (Ω, F, (Ft)t≥0, W, π) be a stochastic basis and let u0 :Ω → V be an F0- measurable random variable. Then there exists a local pathwise solution u to equation (1) up to an Ft-stopping time τ. That is, almost surely, for every t ∈ [0, τ) we have Z t Z t u(t) + [Au(s) + B(u(s)) + F (u(s))] ds = u0 + G(u(s−)) dW (s) 0 0 Z Z + K(u(s−), ξ) dπb(s, ξ) (0,t] E0 Z Z + K(u(s−), ξ) dπ(s, ξ). (0,t] E\E0 Furthermore, P[τ > 0] = 1. Theorem 4.3 will be established in stages in the sections below with the final step in the proof, the piecing-out argument, outlined in Subsection 10.2. In the next subsection we describe the plan for establishing Theorem 4.3. 4.3. The truncated equation. The plan to establish Theorem 4.3 is to establish existence for a simplified version of equation (1) and gradually add back the complexity of the full version. As a first reduction we consider equation (1) with only bounded jumps in the Lévy noise. That is, we consider ( du + [Au + B(u) + F (u)] dt = G(u(t−)) dW (t) + R K(u(t−), ξ) dπ(t, ξ), E0 b u(0) = u0, (83) which is equation (1) in the case where K = 0. We refer to equation (83) as the main SPDE with bounded jump noise. Existence and uniqueness of a local pathwise 5670 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM solution to equation (83) is established in Theorem 9.2. Since the noise process is càdlàg it can only have finitely many jumps above a fixed size in finite time. One can therefore obtain a solution to equation (1) by solving equation (83) up to each jump time, add the jump value to the left-hand limit of the solution at the jump time and then restarting equation (83) up to the next jump time and so on. This is the idea of the piecing-out argument, which is given in Section 10 along with technical details needed to justify “restarting” equation (83). Note, however, that a càdlàg function may have infinitely many small jumps in a finite interval, provided that the sizes of the jumps tend to 0 as they accumulate. For this reason the piecing-out argument does not make the study of SPDEs with Lévy noise a trivial extension of the Wiener noise case. As a second reduction we truncate the B and F terms in the main SPDE with bounded jump noise in order to construct a sequence of processes that exist globally in time and solve equation (83) during an initial period of time. Specifically, we consider  du + [Au + θ(ku − u∗k)B(u) + FR(u)] dt = G(u(t−)) dW (t)   + R K(u(t−), ξ) dπ(t, ξ), E0 b (84) u(0) = u ,  0  8 u0 ∈ L (Ω, F0, P; V ). The truncations that we apply to B and F are defined below. In equation (84), θ : [0, ∞) → [0, 1] is a smooth function with θ ≡ 1 on [0, κ/2] and θ ≡ 0 on [κ, ∞), for some fixed κ > 0 chosen so that 1 κ < , (85) 12c0 where c0 is the constant from inequality (75). Note that θ satisfies tp · θ(t) ≤ κp for every t ≥ 0, p > 0. (86)

The function FR : V → H appearing in (84) is deﬁned by (F (u) if kuk ≤ R FR(u) := R (87) F kuk u if kuk ≥ R.

The function u∗ appearing in (84) is the unique solution to the random linear PDE (with randomness on initial value only)1 ( du + Au dt = 0, ∗ ∗ (88) u∗(0) = u0, (cf. (2.24) in [12]). It is easy to show that for every ﬁxed p ∈ [2, ∞) there exists a p deterministic constant C = C(p) > 0 such that for every u0 ∈ L (Ω; V ) we have

Z t Z t p/2 p p−2 2 2 p sup ku∗(s)k + ku∗(s)k |Au∗(s)| ds + |Au∗(s)| ds ≤ Cku0k , s∈[0,t] 0 0 (89)

1Recall that the solution to the random equation is obtained by ﬁxing ω ﬁrst and then solving as deterministic equation. The uniqueness and existence result to the deterministic version of this equation is classical with initial value in V . Hence, this random PDE has unique solution for every ω. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5671

P-a.s. for all t ∈ [0,T ]. We will apply inequality (89) repeatedly in Subsection n n 5.1 to processes u∗ that are defined as in (88) based on initial conditions u0 that 2 are projections of a fixed u0 ∈ L (Ω; H) onto n-dimensional subspaces of H. In those future applications of (89) it will be very important that the constant C in the upper bound does not depend on n. So, we emphasize that the constant C in inequality (89) does not depend on u0. Existence and uniqueness of a global pathwise solution to (84) is developed in Sections5 to8 with the global existence and uniqueness result stated as Theorem 8.2. Although we have only defined local pathwise existence for the main SPDE (1) in Definition 4.2, the meaning of a global pathwise solution to equation (84) is clear: P-a.s., equation (84) should hold in integral form up to time t for every t < T . In Section9 we remove the truncations from equation (84) and obtain a local pathwise solution to the main SPDE with bounded jump noise. In Section 10 we obtain a local pathwise solution to the main SPDE, equation (1), by incorporating the large jumps of the Lévy noise using the piecing-out argument. The bulk of the argument used to establish the main result, Theorem 4.3, is centered around the truncated equation (84). We will solve the truncated equation (84) using a Galerkin scheme. The Galerkin scheme and estimates are given in Section5. The compactness argument is given in Section6, the passage to the limit in Section7 and the uniqueness argument in Section8. The Skorohod coupling theorem is invoked to pass to the limit. This yields almost sure convergence along a subsequence of Galerkin approximations to a candidate solution for the truncated equation (84) but on a new probability space. Thus, as an intermediate step we use a notion of solution on an a priori unknown probability space. This is known as a martingale solution, which we define below for the truncated equation (84).

Deﬁnition 4.4. A pair (Se, ue) is called a global martingale solution to (84) if 4 ∞ Se = (Ωe, Fe, (Fet)t≥0, Pe ,Wf, πe) is a stochastic basis, if ue ∈ L (Ω; L ([0,T ]; V )) ∩ 4 2 L (Ω;e L ([0,T ]; D(A))), ue is Fet-adapted with càdlàg paths in the H-norm, Pe-a.s., ue(0) is Fe0-measurable with law µ0 satisfying Z 8 kvk dµ0(v) < ∞ (90) V and the equation Z t ue(t) + [Aue(s) + θ(kue(s) − ue∗(s)k)B(ue(s)) + FR(ue(s))] ds 0 Z t Z Z = ue(0) + G(ue(s−)) dWf(s) + K(ue(s−), ξ) dπeb(s, ξ) (91) 0 (0,t] E0 holds in H, Pe-a.s. for all t ∈ [0,T ]. Note that since the probability space is an unknown in the martingale solution problem, it is not possible to specify the initial condition of a martingale solution in advance as a random variable. Instead, we only specify the law of the initial value ue(0) of a martingale solution. We also remark that the moment requirement in (90) 4 ∞ 4 2 and regularity requirement ue ∈ L (Ω; L ([0,T ]; V )) ∩ L (Ω;e L ([0,T ]; D(A))) are wrapped in to Deﬁnition 4.4 for convenience as we analyze the truncated equation (84) here but are not requirements for the general notion of martingale solution. 5672 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Existence of a global martingale solution to the truncated equation (84) is established in Theorem 7.5. By an extension of the Yamada-Watanabe theorem existence of a martingale solution together with pathwise uniqueness yields existence of a unique pathwise solution, i.e., a solution in the sense of Deﬁnition 4.2. Pathwise uniqueness for the truncated equation (84) is established in Proposition 16, which then yields existence of a global pathwise solution to the truncated equation (84) in Theorem 8.2. As mentioned above, the truncations are removed in Section9 to solve equation (83) and the unbounded jump noise is incorporated in Section 10 to solve the main SPDE in equation (1).

5. Galerkin scheme and estimates. In the Galerkin scheme for equation (84) ∞ we use the orthonormal basis (wk)k=1 of H constructed in Subsection 4.1. Recall that each wk belongs to D(A) and is eigenvector of A. We consider the ﬁnite dimensional spaces

Hn := span{w1, . . . , wn}, n ≥ 1.

We denote by Pn the orthogonal projection onto Hn in H. We observe that Pn commutes with A so that PnAv = Av for every v ∈ Hn. We will define the Galerkin system for approximating the truncated equation (84) by projecting the th coefficients in (84) to the space Hn. The n Galerkin approximation to (84) will n be the solution u to the Hn-valued SDE  dun + [Aun + θ(kun − unk)P B(un) + P F (un)] dt  ∗ n n R n R n = PnG(u (t−)) dW (t) + PnK(u (t−), ξ) dπ(t, ξ), (92) E0 b  n n u (0) = Pnu0 =: u0 , n where u∗ is the unique solution to the random ODE ( n n du∗ + Au∗ dt = 0, n n (93) u∗ (0) = u0 . n Note that the functions u∗ satisfy inequality (89) for each p ∈ [2, ∞) with a constant C = C(p) that does not depend on n. n Recall that the function (ω, s, x) 7→ θ(kPn(x) − u∗ (ω, s)k)PnB(Pn(x)) as in (86) satisfies the uniform Lipschitz condition (31), and that FR as in (87) is Lipschitz on Hn, and that G and K satisfy the Lipschitz condition (79). An application of Theorem 3.5 shows that equation (92) has a pathwise unique local solution un up to a positive stopping time σn. We now use the cancellation property (73) and n Proposition6 to show below that the solution u is actually global, i.e., σn = T a.s. (cf. Theorem 3.1 and Lemma 3.2 in [16]). 5.1. Basic estimates. n Proposition 7. Let (u , σn) be the unique maximal local solution to the SDE (92) 2 with initial data u0 ∈ L (Ω, F0, P; H). Then P[σn = T ] = 1. Proof. We apply the Itô formula to un using the function ψ(v) := |v|2 (see Corollary 1) and obtain Z t n 2 n n n n n n |u (t)| + 2 [(u (s), Au (s)) + θ(ku (s) − u∗ (s)k)(PnB(u (s)), u (s))] ds 0 Z t n n = − 2 (u (s),PnFR(u (s))) ds 0 REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5673

Z t n 2 n n + |u0 | + 2 (u (s),PnG(u (s−)) dW (s)) (94) 0 Z Z n n + 2 (u (s),PnK(u (s−), ξ)) dπb(s, ξ) (0,t] E0 Z t Z Z + kP G(un(s))k2 ds + |P K(un(s−), ξ)|2 dπ(s, ξ), n L2(U0,H) n 0 (0,t] E0

n n n 2 P-a.s. for all t ∈ [0, σn). We have (u (s), Au (s)) = ku (s)k by the assumptions on n n the linear term A. By the cancellation property (73) we have (PnB(u (s)), u (s)) ≡ n 0. Since u is Hn-valued and Pn is the orthogonal projection onto Hn in H, we can bound the right-hand side of (94) by dropping each occurrence of Pn. From the observations above we obtain Z t Z t n 2 n 2 n n 2 |u (t)| + 2 ku (s)k ds ≤ |u (s)||FR(u (s))| ds + |u0| 0 0 Z t n n + 2 (u (s),G(u (s−)) dW (s)) 0 Z Z n n + 2 (u (s),K(u (s−), ξ)) dπb(s, ξ) (95) (0,t] E0 Z t Z Z + kG(un(s))k2 ds + |K(un(s−), ξ)|2 dπ(s, ξ), L2(U0,H) 0 (0,t] E0

P-a.s. for all t ∈ [0, σn). From inequality (95) we obtain

h i Z t∧σn E sup |un(s)|2 + 2E kun(s)k2 ds s∈[0,t∧σn) 0 Z t∧σn n n 2 ≤ E |u (s)||FR(u (s))| ds + E|u0| 0 0 h Z t i + 2E sup (un(s),G(un(s−)) dW (s)) 0 t ∈[0,t∧σn) 0 (96) h Z Z i + 2E sup (un(s),K(un(s−), ξ)) dπ(s, ξ) 0 0 b t ∈[0,t∧σn) (0,t ] E0 Z t∧σn + E kG(un(s))k2 ds L2(U0,H) 0 Z Z + E |K(un(s−), ξ)|2 dν(ξ) ds. (0,t∧σn] E0 From this point we can apply the BDG inequality, and use the growth conditions (77a) and (78) in a similar way to the proof of Proposition5 to deduce that there exists a constant C = C(R,T ) > 0, which does not depend on n, such that

Z t∧σn h n 2i n 2 2 E sup |u (s)| + ku (s)k ds ≤ C(1 + E|u0| ) s∈[0,t∧σn) 0 Z t " # + C E sup |un(r)|2 ds. 0 r∈[0,s∧σn) (97) 5674 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Applying the deterministic Gronwall inequality to (97), we ﬁnd that there exists a larger constant Ce > 0, which does not depend on n, such that h n 2i 2 E sup |u (s)| ≤ Ce(1 + E|u0| ), (98) s∈[0,t∧σn) for all t ∈ [0,T ]. By taking t := T we see that P[σn = T ] = 1 by Proposition6. According to the construction of local solutions in Theorem 3.5, the process un solves equation (92) on the interval [0,T ) a.s. Of course, we may replace T by T +1 at the beginning to reach the conclusion that un solves equation (92) on the closed n interval [0,T ] and that u is càdlàg in Hn on [0,T ] a.s. We now record immediate consequences of the inequalities (97) and (98). Corollary 2. Let un be the solution to the SDE (92) for a given cutoﬀ parameter 2 R > 0 and with initial condition u0 ∈ L (Ω, F0, P; H). Then there exists a constant C = C(R,T ) > 0 such that

h n 2i 2 sup E sup |u (s)| ≤ C(1 + E|u0| ), (99) n≥1 s∈[0,T ] and Z T h n 2 i 2 sup E ku (s)k ds ≤ C(1 + E|u0| ). (100) n≥1 0 n ∞ We are ready to make the main estimates on the solutions (u )n=1 to (92) for each value of the cutoff parameter R > 0. The estimates given below in Proposition 8 are analogous to Lemma 3.1 of [12]. The difference between our setting and stochastic fluid model considered in [12] is the presence of the Lévy noise term. We will focus our attention on the new terms that arise from the Lévy noise and will refer to [12] for the details involved in estimating the remaining terms. Proposition 8. Let un be the solution to the SDE (92) for a given cutoff parameter 8 R > 0 as in (87) with initial data u0 ∈ L (Ω, F0, P; V ). Then there exists a positive constant C = C(R,T ) such that

h n 4i 4 sup E sup ku (s)k ≤ C(1 + Eku0k ), (101) n≥1 s∈[0,T ] Z T n 2 n 2 8 sup E ku (s)k |Au (s)| ds ≤ C(1 + Eku0k ), (102) n≥1 0 and Z T 2 n 2 8 sup E |Au (s)| ds ≤ C(1 + Eku0k ). (103) n≥1 0 n n n Proof. For each positive integer n, deﬁne v (s) := u (s) − u∗ (s) for all s ∈ [0,T ]. Then vn satisﬁes the equation  dvn + [Avn + θ(kvnk)P B(un) + P F (un)] dt  n n R n R n = PnG(u (t−)) dW (t) + PnK(u (t−), ξ) dπ(t, ξ), (104) E0 b vn(0) = 0.

1/2 We apply the linear operator A , which is bounded on Hn, to equation (104) and then apply the Itô formula using the function ψ(v) := |v|4 up to the stopping time n n τm := inf{t > 0 : kv (t)k > m} ∧ T. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5675

The expression that we obtain from the Itô formula is rather long (cf. equation (47)). We ﬁnd that the expression

n Z t∧τm 1/2 n 2 1/2 n 1/2 n R2(t) := 4 |A v (s)| (A v (s),A PnG(u (s−)) dW (s)) 0 t∧τ n Z m 2 1/2 n 2 1/2 n + 2 |A v (s)| A PnG(u (s−)) ds 0 L2(U0,H) n Z t∧τm 1/2 n ∗ 1/2 n 2 + 4 [A PnG(u (s−))] A v (s) U ds 0 0 Z Z 1/2 n 2 1/2 n 1/2 n + 4|A v (s−)| (A v (s−),A PnK(u (s−), ξ))H dπ(s, ξ) n b (0,t∧τm] E0 Z Z 1/2 n 1/2 n 4 1/2 n 4 + [|A v (s−) + A PnK(u (s−), ξ)| − |A v (s−)| ] dπ(s, ξ) n (0,t∧τm] E0 Z Z 1/2 n 2 1/2 n 1/2 n − 4|A v (s−)| (A v (s−),A PnK(u (s−), ξ))H dπ(s, ξ), n (0,t∧τm] E0 P-a.s. for all t ∈ [0,T ]. Using the fact that (A1/2u, A1/2v) = ((u, v)) for all u, v ∈ Hn, the equality R1(t) = R2(t) above reduces to n Z t∧τm n n 4 n 2 n 2 kv (t ∧ τm)k + 4 kv (s)k |Av (s)| ds 0 n Z t∧τm n 2 n n n + 4 kv (s)k θ(kv (s)k)(PnB(u (s)), Av (s)) ds 0 n Z t∧τm n 2 n n + 4 kv (s)k (Av (s),PnFR(u (s)))] ds 0 n Z t∧τm n 2 n n = 4 kv (s)k ((v (s),PnG(u (s−)) dW (s))) 0 t∧τ n Z m h i + 2kvn(s)k2 kP G(un(s−))k2 + 4|[P G(un(s−))]∗Avn(s)|2 ds n L2(U0,V ) n U0 0 Z Z n 2 n n + 4kv (s−)k ((v (s−),PnK(u (s−), ξ))) dπ(s, ξ) n b (0,t∧τm] E0 Z Z n n 4 n 4 + kv (s−) + PnK(u (s−), ξ)k − kv (s−)k dπ(s, ξ) n (0,t∧τm] E0 Z Z n 2 n n − 4kv (s−)k ((v (s−),PnK(u (s−), ξ))) dπ(s, ξ) n (0,t∧τm] E0 5676 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

n n n n n =: I1(t ∧ τm) + I2(t ∧ τm) + I3(t ∧ τm) + I4(t ∧ τm) − I5(t ∧ τm). (105) We move the B and F terms from the right-hand side of (105) to the left-hand side, then take absolute value on both sides. The B and F terms on the left-hand side of (105), as well as the Wiener integral and Riemann integral on the right-hand side of (105) can be estimated in exactly the same way as in the proof of Lemma 3.1 in [12]. Using inequality (75), the choice (85) of κ and the growth conditions (77a) and (78) we ﬁnd that there exists a constant C = C(R) > 0 that depends on the cutoﬀ parameter R but not on n such that the following estimates hold: Z t n 2 n n n 4 kv (s)k θ(kv (s)k)|(PnB(u (s)), Av (s))| ds 0 (106) Z t n 2 n 2 n 2 n 2 ≤ kv (s)k |Av (s)| + Cku∗ (s)k |Au∗ (s)| ds, 0 Z t Z t n 2 n n n 2 n 2 4 kv (s)k |(Av (s),PnFR(u (s)))| ds ≤ kv (s)k |Av (s)| ds 0 0 Z t n 4 n 4 + C(1 + ku∗ (s)k + kv (s)k ) ds, 0 (107) " # 1 h n 4i E sup |I1(r)| ≤ E sup kv (s)k r∈[0,t∧τ n ] 4 s∈[0,t∧τ n ] m m (108) n Z t∧τm n 4 n 4 + CE 1 + ku∗ (s)k + kv (s)k ds, 0 and " # n Z t∧τm n 4 n 4 E sup |I2(r)| ≤ CE 1 + ku∗ (s)k + kv (s)k ds. (109) n r∈[0,t∧τm] 0

We now give detailed estimates of the terms I3(t) and I4(t) − I5(t) in (105). We h i estimate E sup n |I3(r)| using the BDG inequality (Proposition2) and rea- r∈[0,t∧τm] soning as in the proof of Proposition5 (cf. inequality (54)). We find that there exists a constant C > 0 that does not depend on n (and may change on each line) such that " # E sup |I3(r)| n r∈[0,t∧τm] Z Z 1/2 n 4 n n 2 ≤ CE kv (s−)k · |((v (s−),PnK(u (s−), ξ)))| dπ(s, ξ) n (0,t∧τm] E0 Z Z 1/2 ≤ CE kvn(s−)k6kK(un(s−), ξ)k2 dπ(s, ξ) n (0,t∧τm] E0 h Z Z 1/2i ≤ CE sup kvn(s)k2 kvn(s−)k2kK(un(s−), ξ)k2 dπ . n n s∈[0,t∧τm] (0,t∧τm] E0 Now we use Schwarz’s inequality to deduce that there exists a constant C > 0, which does not depend on n or m, such that " # 1 h n 4i E sup |I3(s)| ≤ E sup kv (s)k n 4 n s∈[0,t∧τm] s∈[0,t∧τm] REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5677 Z Z + CE kvn(s−)k2kK(un(s−), ξ)k2 dπ(s, ξ). n (0,t∧τm] E0 Reasoning as in the proof of Proposition5, we find that " # 1 h n 4i E sup |I3(s)| ≤ E sup kv (s)k s∈[0,t∧τ n ] 4 s∈[0,t∧τ n ] m m (110) n Z t∧τm n 4 n 4 + CE 1 + ku∗ (s)k + kv (s)k ds. 0 h i We estimate E sup n |I4(s) − I5(s)| using the inequality (55) and a similar s∈[0,t∧τm] reasoning to the proof of Proposition5 (cf. inequality (56)). We find that there exists a constant C > 0 that does not depend on n or m such that E sup |I4(s) − I5(s)| n s∈[0,t∧τm] Z Z n n 4 n 4 = E kv (s−) + PnK(u (s−), ξ)k − kv (s−)k n (0,t∧τm] E0

n 2 n n − 4kv (s−)k ((v (s−),PnK(u (s−), ξ))) dπ(s, ξ) Z Z n 2 n 2 n 4 ≤ CE kv (s−)k · kPnK(u (s−), ξ)k + kPnK(u (s−), ξ)k dπ(s, ξ) n (0,t∧τm] E0 Z Z ≤ CE kvn(s−)k2 · kK(un(s−), ξ)k2 + kK(un(s−), ξ)k4 dν(ξ) ds. n (0,t∧τm] E0 We estimate the ﬁrst term in the integrand above using (78) for the space V and we estimate the second term in the integrand above using inequality (80). These estimates and similar reasoning to the proof of Proposition5 (cf. inequality (56)) lead to " # n Z t∧τm n 4 n 4 E sup |I4(s) − I5(s)| ≤ CE 1 + ku∗ (s)k + kv (s)k ds. (111) n s∈[0,t∧τm] 0 We are now ready to return to (105) and combine the estimates above. By combining the inequalities (106), (107), (108), (109), (110) and (111) we ﬁnd that there exists a constant C = C(R,T ) > 0 such that

" # n Z t∧τm E sup kvn(s)k4 + kvn(s)k2|Avn(s)|2 ds n s∈[0,t∧τm] 0 Z t " ! # n 2 n 2 n 4 n 4 ≤ C E 1 + ku∗ (s)k |Au∗ (s)| + sup kv (r)k + ku∗ (s)k ds, n 0 r∈[0,s∧τm] (112) for all t ∈ [0,T ] and for all positive integers n and m. By applying the deterministic n Gronwall inequality and using the estimate (89) on u∗ we conclude that there exists a constant C = C(R,T ) > 0 that does not depend on n or m such that

" # n Z T ∧τm n 4 n 2 n 2 4 E sup kv (s)k + E kv (s)k |Av (s)| ds ≤ C(1 + Eku0k ). (113) n s∈[0,T ∧τm] 0 5678 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

n Since τm ↑ T as m → ∞, we ﬁnd that

" # Z T n 4 n 2 n 2 4 E sup kv (s)k + E kv (s)k |Av (s)| ds ≤ C(1 + Eku0k ). (114) s∈[0,T ] 0

We deduce (101) by combining (114) and (89). It is easy to see that the estimate (102) follows from (101) and (103) by an application of Young’s inequality. So, it remains only to establish (103). Using (89) we see that (103) follows from

Z T 2 n 2 8 sup E |Av (s)| ds ≤ C(1 + Eku0k ). (115) n≥1 0

In order to establish (115) we apply A1/2 to equation (104) and then apply the Itô formula using the function ψ(v) := |v|2 (cf. Corollary1). We ﬁnd that

Z t n 2 n 2 n n n kv (t)k + 2 [|Av (s)| + θ(kv (s)k)(PnB(u (s)), Av (s))] ds 0 Z t n n + (Av (s),PnFR(u (s))) ds 0 Z t Z t = 2 ((vn(s),P G(un(s−)) dW (s))) + kP G(un(s−))k2 ds n n L2(U0,V ) 0 0 Z Z n n + 2((v (s−),PnK(u (s−), ξ))) dπb(s, ξ) (0,t] E0 Z Z n 2 + kPnK(u (s−), ξ)k dπ(s, ξ) (0,t] E0

=: J1(t) + J2(t) + J3(t) + J4(t), (116)

P-a.s. for all t ∈ [0,T ]. As in Lemma 3.1 in [12], estimates for the B, F , Wiener and Riemann integral terms in (116) can be obtained in an analogous way to estimates (106), (107), (108) and (109). We have:

Z t n n n −2 θ(kv (s)k)(PnB(u (s)), Av (s)) ds 0 (117) Z t h −1/2 n 2 n 2 n 2i ≤ 2 |Av (s)| + Cku∗ (s)k |Au∗ (s)| ds, 0

Z t n 2 n n −4 kv (s)k (Av (s),PnFR(u (s))) ds 0 (118) Z t h −1/2 n 2 n 2 n 2 i ≤ 2 |Av (s)| + C(1 + ku∗ (s)k + kv (s)k ) ds, 0

Z t 2 n 4 n 4 E |J1(t)| ≤ CE 1 + ku∗ (s)k + kv (s)k ds, (119) 0

Z t n 2 n 2 E [|J2(t)|] ≤ CE 1 + ku∗ (s)k + kv (s)k ds. (120) 0 REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5679

2 We rewrite E[|J3(t)| ] using the isometric formula (21) and then use (79) to obtain the estimate Z t Z 2 n n 2 E[|J3(t)| ] = 4E |((v (s−),K(u (s−), ξ)))| dν(ξ) ds 0 E0 Z t Z ≤ 4E kvn(s−)k2kK(un(s−), ξ)k2 dν(ξ) ds 0 E0 Z t ≤ CE kvn(s)k2(1 + kun(s)k2) ds 0 Z t n 4 n 4 ≤ CE 1 + ku∗ (s)k + kv (s)k ds, (121) 0 where the constant C does not depend on n. Before turning to the term J4(t), n 2 we observe that the function f(s, ξ) := kPnK(u (s−), ξ)k belongs to the space 1 2 1 Fν,T (H) ∩ Fν,T (H). Indeed, the fact that f ∈ Fν,T (H) follows from the growth n 2 2 condition (78) and the fact that u ∈ L (Ω × [0,T ]; V ). The fact that f ∈ Fν,T (H) follows from the growth condition (80) and the fact that un ∈ L4(Ω × [0,T ]; V ) 1 2 (according to (101)). Since f ∈ Fν,T (H) ∩ Fν,T (H) we have Z Z Z t Z n 2 n 2 J4(t) = kPnK(u (s−), ξ)k dπb(s, ξ) + kPnK(u (s−), ξ)k dν(ξ) ds (0,t] E0 0 E0 (122) (cf. Theorem 2.16). Using the isometric formula (21) and the growth conditions (78) and (80) we ﬁnd that Z t Z 2 n 4 E[|J4(t)| ] ≤ 2E kPnK(u (s−), ξ)k dν(ξ) ds 0 E0 Z t Z 2 n 2 + 2E kPnK(u (s−), ξ)k dν(ξ) ds 0 E0 Z t Z t 2 ≤ CE (1 + kun(s)k4) ds + CE (1 + kun(s)k2) ds 0 0 Z t n 4 n 4 ≤ CE [1 + ku∗ (s)k + kv (s)k ] ds, (123) 0 for a constant C that depends on T but does not depend on n. Returning to (116) at time t = T , we drop the term kvn(T )k2 from the left-hand side and take the absolute value squared on both sides. After combining the estimates (117), (118), (119), (120), (121) and (123) we obtain

Z T 2 Z T 2 n 2 n 2 n 2 E |Av (s)| ds ≤ CE ku∗ (s)k |Au (s)| ds 0 0 (124) Z T n 4 n 4 + CE 1 + ku∗ (s)k + kv (s)k ds. 0 We now obtain (115) by combining (124) with (89) and (101). As noted above, (115) and (124) imply (103) and (102).

n n n We will continue to use the notation v := u − u∗ that was introduced in the proof of Proposition8. 5680 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

5.2. Estimates in fractional Sobolev spaces.

Deﬁnition 5.1. We recall that if X is a real, separable Hilbert space, if 1 < p < ∞ and α ∈ (0, 1), the space

The next two results are about continuity of the stochastic integration maps, with respect to W and πb, into fractional Sobolev spaces using the space H in the role of X. In order to state these results it is convenient to introduce the shorthand notation Z t Z Z W πb It (Ψ) := Ψ(s) dW (s) and It (f) := f(s, ξ) dπb(s, ξ), 0 (0,t] E0 for Ψ ∈ L2 (H), f ∈ F2 (H) and t ∈ [0,T ]. A special case of Lemma 2.1 in [17] U0,T ν,T says that IW is continuous as a mapping from the space L2 (H) into the space U0,T L2(Ω; W α,2(0,T ; H)) for all 0 < α < 1/2. The precise statement of the result is below.

Lemma 5.2. For every α ∈ (0, 1/2), there exists a constant C = C(α, T ) > 0 such that for every Ψ ∈ L2 (H) we have U0,T

T 2 Z E IW (Ψ) ≤ CE kΨ(s)k2 ds. (125) W α,2(0,T ;H) L2(U0,H) 0 A similar continuity result holds for the stochastic integral with respect to a πb 2 compensated Poisson random measure. Namely, I maps Fν,T (H) continuously into L2(Ω; W α,2(0,T ; H)). That is, there exists a constant C = C(α, T ) > 0 such 2 that for every f ∈ Fν,T (H) we have

T 2 Z Z πb 2 E I (f) W α,2(0,T ;H) ≤ CE |f(s, ξ)|H dν(ξ) ds. (126) 0 E We refer to the appendix of [9] for a proof of this result. With these preliminaries in hand we are ready to prove that the Galerkin ap- n ∞ proximations (u )n=1 to the truncated equation (84) are bounded in the space L2(Ω; W α,2([0,T ]; H)) for each α ∈ (0, 1/2).

Proposition 9. Let un be the solution to the SDE (92) for a given cutoﬀ parameter 8 R > 0 with initial data u0 ∈ L (Ω, F0, P; V ). Then for every α ∈ (0, 1/2) there exists a constant C = C(α, R, T ) > 0 such that

n 2 sup E ku kW α,2([0,T ];H) ≤ C. (127) n≥1 REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5681

Proof. By inequality (125) and the growth condition (78), we have

T 2 Z E IW (P G(un)) ≤ CE kP G(un(s−))k2 ds n W α,2([0,T ];H) n L2(U0,H) 0 (128) Z T ≤ CE (1 + |un(s)|2) ds 0 for some constant C that does not depend on n. By (101) the right-hand side of (128) is bounded by a constant, independently of n. Similarly, (126) and the growth condition (78) give Z T Z 2 2 πb n n E I (PnK(u , ·)) W α,2([0,T ];H) . E kPnK(u (s−), ξ)kH dν(ξ) ds 0 E0 Z T n 2 . E (1 + |u (s)| ) ds. 0 By (101) the right-hand side of this inequality is bounded by a constant that depends on T but not on n. In light of these estimates it is suﬃcient to prove that 2 n W n πb n sup E u − I (PnG(u )) − I (PnK(u , ·)) W α,2([0,T ];H) < ∞. (129) n≥1

n n W n π n Let f := u − I (PnG(u )) − I b(PnK(u , ·)). Then we have

n 2 kf kW α,2([0,T ];H) Z · Z · Z · 2 n n n n n = u0 − Au ds − θ(kv k)PnB(u ) ds − PnFR(u ) ds 0 0 0 W α,2([0,T ];H) Z · Z · Z · 2 n n n n n . u0 − Au ds − θ(kv k)PnB(u ) ds − PnFR(u ) ds 0 0 0 W 1,2([0,T ];H) Z · Z · Z · 2 n n n n n . u0 − Au ds − θ(kv k)PnB(u ) ds − PnFR(u ) ds 0 0 0 L2([0,T ];H) Z T Z T Z T n 2 n 2 n 2 n 2 + |Au | ds + θ(kv k) |B(u )| ds + |FR(u )| ds, 0 0 0 Z T Z T Z T 2 n 2 n 2 n 2 n 2 . |u0| + |Au | ds + θ(kv k) |B(u )| ds + |FR(u )| ds 0 0 0 2 =: |u0| + I1 + I2 + I3, (130) where the hidden constant depends on T but not on n. By inequality (103) there exists a constant C = C(T ) > 0, independent of n, such that

E[I1] ≤ C. (131)

We estimate I2 using using inequality (75) and the cutoﬀ property (86) (in a similar way to the proof of Lemma 3.1 in [12]) and obtain Z T n 2 n 2 n n 2 E[I2] ≤ CE θ(kv (s)k) [|B(v (s))| + |B(v (s), u∗ (s))| ] ds 0 Z T n 2 n n 2 n 2 + CE θ(kv (s)k) [|B(u∗ (s), v (s))| + |B(u∗ (s))| ] ds 0 5682 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Z T n 2 n 2 n 2 n 2 n 2 ≤ CE θ(kv (s)k) [kv (s)k |Av (s)| + ku∗ (s)k |Au∗ (s)| ] ds, 0 Z T n 2 n 2 n 2 ≤ CE [κ|Av (s)| + ku∗ (s)k |Au∗ (s)| ] ds, (132) 0 where C is a constant that does not depend on n. By (115) and (89) we see that the right-hand side of (132) is bounded by a constant that depends on T but not on n. We estimate I3 using (77a) and inequality (101) and we ﬁnd that there exists a constant C = C(R,T ) > 0, independent of n, such that Z T n 2 E[I3] ≤ CE (1 + ku (s)k ) ds ≤ C. (133) 0 Combining estimates (131), (132) and (133) in (130), we deduce (127).

6. Tightness. We have finished with the estimates on the Galerkin approximations and now we are ready to establish compactness properties. Because of the discontinuity of the Lévy noise, classical compactness results of fractional Sobolev spaces into spaces of continuous functions (e.g., Theorem 2.2 of [17]) are not avail- able in the present setting. As a replacement we will establish a certain compactness property of the laws of the Galerkin approximations known as tightness. We now recall the definition of tightness. Definition 6.1. A set Π of Borel probability measures on a metric space S is said to be tight if for every ε > 0 there exists a compact set K ⊆ S such that µ(Kc) < ε for every µ ∈ Π. Prohorov’s theorem states that a family Π of Borel probability measures on a complete, separable metric space is tight if and only if the closure of Π in the topology of weak convergence of probability measures is compact (see, e.g., Theorems 5.1 and 5.2 in [4]). In this section we show that the laws of the Galerkin approximations n ∞ (u )n=1 to the truncated equation (84) are tight on various spaces. We will then obtain the law of a candidate solution to equation (84) by passing to a weakly con- n ∞ vergent subsequence of the laws of (u )n=1. We use this method, in part, to ensure that the candidate solution will have càdlàg sample paths in the H-norm a.s. To n ∞ be specific, we will show in Proposition 11 that the laws of (u )n=1 are tight on the space D([0,T ]; H) of càdlàg functions from [0,T ] → H endowed with the Skorohod topology. See, e.g., page 116 in [15] or page 123 in [4] for the definition of the Skoro- hod topology. Rather than defining the Skorohod topology, we are content to recall the following facts: the Skorohod topology is weaker than the topology of uniform convergence on D([0,T ]; H) and the Skorohod topology is separable and metrizable by a complete metric, while the topology of uniform convergence on D([0,T ]; H) is not separable. The setting of complete, separable metric spaces is more tractable in the theory of weak convergence of probability measures (cf. Prohorov’s theorem above) and this the reason for endowing D([0,T ]; H) with the Skorohod topology. It can be difficult to deduce tightness of a sequence of probability measures on D([0,T ]; H) by verifying Definition 6.1 directly. Instead, it is common to verify a sufficient condition for tightness introduced by Aldous in [1]. The version of the Aldous condition that we state below as Proposition 10 and apply in Proposition 11 is a special case of Theorem 3.2 in [27]. ∞ Proposition 10. Let (Yn)n=1 be a sequence of H-valued processes defined on a filtered probability space (Ω, F, (Ft)t≥0, P), such that each Yn is Ft-adapted and has REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5683 càdlàg sample paths, P-a.s. Then for each fixed T > 0 the following two conditions ∞ are sufficient for the laws of (Yn)n=1 to form a tight family of probability measures on D([0,T ]; H) endowed with the Skorohod topology: ∞ i) the laws of the random variables (Yn(t))n=1 form a tight family of probability measures on H, for all t in a dense subset of [0,T ], and ii) for every ε > 0 and η > 0 there exists δ > 0 such that for every sequence ∞ (τn)n=1 of Ft-stopping times that are bounded by T , a.s., one has

sup sup P[|Yn((τn + t) ∧ T ) − Yn(τn)| ≥ η] ≤ ε. (134) n≥1 t∈[0,δ] For comparison, we mention that Theorem 13.2 of [38] is a deterministic analogue of the Aldous condition. n ∞ We will apply Proposition 10 to the laws of the sequence (u )n=1 in two steps: ﬁrst, by verifying condition i) in Lemma 6.2 and, second, by verifying condition ii) in Proposition 11 below. Lemma 6.2. Let un be the solution to the SDE (92) for a given cutoﬀ parameter 8 R > 0 with initial data u0 ∈ L (Ω, F0, P; V ). Then for every t ∈ [0,T ], the laws of n ∞ the H-valued random variables (u (t))n=1 are tight.

Proof. For each r > 0 the closed ball Br := {v ∈ V : kvk ≤ r} is a compact subset of H. By Chebyshev’s inequality and inequality (101) we have 1 C(1 + Eku k4) P[un(t) 6∈ B ] ≤ E[kun(t)k2] ≤ 0 r r2 r2 for all t ∈ [0,T ], where C > 0 is independent of n and t. This shows that the laws n ∞ of (u (t))n=1 are tight on H. Proposition 11. Let un be the solution to the SDE (92) for a given cutoff param- 8 n ∞ eter R > 0 with initial data u0 ∈ L (Ω, F0, P; V ). Then the laws of (u )n=1 form a tight sequence of probability measures on D([0,T ]; H) endowed with the Skorohod topology. Proof. We have verified condition i) of Proposition 10 in Lemma 6.2. In order to verify condition ii) it is sufficient, by Chebyshev’s inequality, to show that n n 2 lim sup sup E|u ((τn + t) ∧ T ) − u (τn)| = 0. (135) s→0+ n≥1 t∈[0,s] Let δ > 0 and let n be a positive integer. It is easy to see that

We need to show that

lim sup sup E[Ji] = 0 (136) δ→0+ n∈N t∈[0,δ] holds for i = 1, 2, 3, 4, 5. This is clear for i = 1 because of inequality (103). We estimate J2 as in inequality (132) and obtain Z T n 2 n n 2 E[J2] ≤ tCE [κ|Au (s)| + ku∗ (s)k|Au (s)| ] ds ≤ tC, 0 where the constant C does not depend on n or t. From the inequality above we see that (136) holds for i = 2. We estimate J3 as in inequality (133) and obtain

2 E[J3] ≤ tC(1 + E[ku0k ]), which implies that (136) holds when i = 3. For J4 we use the isometric formula (21) and the growth condition (78) to obtain

h Z (τn+t)∧T i E[J ] = E kP G(un(s−))k2 ds 4 n L2(U0,H) τn h Z (τn+t)∧T i ≤ CE (1 + |un(s)|2) ds τn h i ≤ CtE 1 + sup |un(s)|2 . t∈[0,T ]

Using Corollary2 we see that (136) holds for i = 4. Similarly, for J5 we have

Z (τn+t)∧T Z h n 2 i E[J5] = E |PnK(u (s−), ξ)| dν(ξ) ds τn E0 h Z (τn+t)∧T i ≤ CE (1 + |un(s)|2) ds τn h i ≤ CtE 1 + sup |un(s)|2 , s∈[0,T ] and it follows from Corollary2 that (136) holds for i = 5. We conclude that (135) n ∞ holds, so the Aldous condition (134) is satisﬁed. This shows that the laws of (u )n=1 are tight in D([0,T ]; H) endowed with the Skorohod topology.

We now recall a deterministic compactness result that we will use to show that n ∞ 2 the laws of (u )n=1 are tight on L ([0,T ]; V ). Since D(A) is compactly embedded in V , which is continuously embedded in H, we have L2([0,T ]; D(A)) ∩ W α,2([0,T ]; H) ⊂⊂ L2([0,T ]; V ); see Theorem 2.1 of [17] for instance (cf. inequalities (103) and (127)). Using the compactness result above and Proposition9 it is straightforward to show that ∞ the Galerkin approximations (un)n=1 of the truncated equation (84) are tight as L2([0,T ]; V )-valued random variables.

Proposition 12. Let un be the solution to the SDE (92) for a given cutoﬀ param- 8 n ∞ eter R > 0 with initial data u0 ∈ L (Ω, F0, P; V ). Then the laws of (u )n=1 are tight on L2([0,T ]; V ). REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5685

Proof. For r > 0 let 2 1/4,2 Br := u ∈ L ([0,T ]; D(A)) ∩ W ([0,T ]; H):

kukL2([0,T ];D(A)) + kukW 1/4,2([0,T ];H) ≤ r . 2 Then Br is a compact subset of L ([0,T ]; V ). Using Chebyshev’s inequality we see that n n n P[u 6∈ Br] ≤ P[ku kL2([0,T ];D(A)) > r/2] + P[ku kW 1/4,2([0,T ];H) > r/2] 4 h i ≤ E kunk2 + kunk2 . r2 L2([0,T ];D(A)) W 1/4,2([0,T ];H) By (103) and Proposition9 there exists a constant C = C(T,R) > 0 such that n 2 P[u 6∈ Br] ≤ C/r for every positive integer n. This shows that the laws of n ∞ 2 (u )n=1 are tight on L ([0,T ]; V ). n ∞ We can combine Propositions 11 and 12 to show that the laws of (u )n=1 are tight on 2 Xu := D([0,T ]; H) ∩ L ([0,T ]; V ). (137) In preparation for applying the theory of weak convergence for probability measures on Xu we observe that the space Xu is metrizable by a complete, separable metric; see, e.g., [9] for details. Proposition 13. Let un be the solution to the SDE (92) for a given cutoﬀ param- 8 n ∞ eter R > 0 with initial data u0 ∈ L (Ω, F0, P; V ). Then the laws of (u )n=1 are tight on Xu.

Proof. For every ε > 0 there exist compact sets K1 ⊂ D([0,T ]; H) and K2 ⊂ L2([0,T ]; V ) such that n sup P[u 6∈ Ki] ≤ ε/2 n∈N for i = 1, 2. It is easy to see that K1 ∩ K2 is sequentially compact in Xu and that n sup P[u 6∈ K1 ∩ K2] ≤ ε. n≥1 n ∞ This shows that the laws of (u )n=1 are tight on Xu. 7. Martingale solution to the truncated equation. We have defined the Galerkin solutions (92) to the truncated equation (84). The Galerkin solutions satisfy the a priori estimates (101), (103) and (127), which show that the laws of ∞ 2 (un)n=1 are tight in L (0,T ; V ) (see Proposition 12). We have also verified the ∞ Aldous condition in Proposition 11 to show that the laws of (un)n=1 are tight on ∞ D([0,T ]; H), so we can extract from (un)n=1 a subsequence whose laws converge weakly to some random variable u that takes value in the space Xu := D([0,T ]; H)∩ L2(0,T ; V ). Our next goal is to show that u is a global pathwise solution to the truncated equation (84). However, the weak convergence2 un ⇒ u in law and the weak convergence along a further subsequence in L2(Ω; L2([0,T ]; D(A))) that is provided by (103) will not be enough to establish (84) by passing to the limit in (92). Instead, we will use the Skorohod coupling theorem to upgrade the weak n n n D convergence u ⇒ u to a.s. convergence ue → ue in Xu for random variables ue = un D and ue = u that are defined on a new probability space (Ωe, Fe, Pe). In this section we will make the application of the Skorohod coupling theorem precise and show that

2Weak convergence of laws is usually denoted by un ⇒ u. 5686 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM ue is a martingale solution to the truncated equation (84); i.e., that ue satisﬁes (91). In Subsection 7.2 we will return to our goal of proving that u satisﬁes equation (84) in integral form.

7.1. Convergence lemmas. In preparation for applying the Skorohod coupling theorem we mention that the law of the Wiener process W is a Borel probability measure on the complete, separable metric space C([0,T ]; U). Every Borel probability measure on a complete, separable metric space is tight (see, e.g., Theorem 1.3 in [4]), so the law of W is tight. The Poisson random measure π takes val- #∗ ues in the space N[0,∞)×E of counting measures on (0, ∞) × E that are finite on bounded sets. Under the topology of weak-# convergence (i.e., weak convergence of measures with test functions that are bounded, continuous and have bounded #∗ support) the space N[0,∞)×E is separable and metrizable by a complete metric (see, #∗ e.g., Proposition 9.1.IV. in [10]). Therefore, the law of π is tight on N[0,∞)×E. It n n ∞ is clear that u0 := Pnu0 → u0, P-a.s. in V , in particular, the laws of (u0 )n=1 are tight on V and their weak limit is the measure µ0(·) = P[u0 ∈ · ]. In Proposition n ∞ 13 we showed that the laws of (u )n=1 are tight on Xu. It is easy to combine the n n ∞ tightness results listed above to show that the laws of ((u0 , u , W, π))n=1 are tight on the space #∗ X := V × Xu × C([0,T ]; U) × N[0,∞)×E , which is endowed with the product topology. Since each set in this Cartesian product is separable and metrizable by a complete metric, so is X . By Prohorov’s n n ∞ Theorem there exists a subsequence of ((u0 , u , W, π))n=1 indexed by some infinite n n set Λ ⊆ N, say, such that the laws of ((u0 , u , W, π))n∈Λ converge weakly to a Borel probability measure µ on X . It is clear that the marginal of µ in the first coordinate is the measure µ0 on V . The measure µ gives us a candidate for a martingale solution to the truncated equation (84) via the Skorohod coupling theorem, which we state and apply next.

Proposition 14. Suppose that u0 has law µ0 that satisﬁes (90). Then there exists a ﬁltered probability space (Ωe, Fe, (Fet)t≥0, Pe) and X -valued random variables n n (u0, u, W,f π) and (u0 , u , Wfn, πn) such that e e e e e e n∈Λ

n n D n n i) (ue0 , ue , Wfn, πen) = (u0 , u , W, π) for every n ∈ Λ, (ue0, u,e W,f πe) has law µ, ue0 has law µ0 and n n (ue0 , ue , Wfn, πen) → (ue0, u,e W,f πe) in X , Pe-a.s., as n → ∞ along Λ, ii) for every n ∈ Λ we have (Wfn, πen) = (W,f πe) everywhere on Ωe and n iii) for every n ∈ Λ, ue is a global martingale solution to (92) with respect to the n n n stochastic basis (Ωe, Fe, (Fet)t≥0, Pe, Wf, πe) and ue (0) = ue0 Pe-a.s., i.e., ue is an n Fet-adapted process with càdlàg paths in H, Pe-a.s., ue0 is Fe0-measurable and Z t Z t Z t n n n n n n ue (t) + Aue (s) ds + θ(kue (s) − ue∗ (s)k)PnB(ue (s)) ds + PnFR(ue (s)) ds 0 0 0 Z t Z Z n n n = ue0 + PnG(ue (s)) dWf(s) + PnK(ue (s−), ξ) dπeb(s, ξ), 0 (0,t] E0 (138) REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5687

n holds in Hn, Pe-a.s., for all t ∈ [0,T ], where ue∗ solves the random ODE ( n n due∗ + Aue∗ dt = 0, n n (139) ue∗ (0) = ue0 . Proof. Part i) is just the Skorohod coupling theorem along with the fact that the marginal of µ in its ﬁrst coordinate is µ0. Part ii) can be arranged because the n n same noise (W, π) is used for each n in the original sequence ((u0 , u , W, π))n∈Λ; see Theorem C.1 in [5]. Part iii) in the Wiener noise case, i.e., with K ≡ 0, was established by Bensoussan in [3]. A proof of part iii) in the Lévy noise and in inﬁnite dimension can be found in the appendix of [9].

n n n As in Sections5 and6 we will use the notation ve (s) := ue (s) − ue∗ (s). The remainder of this section is devoted to showing that ue solves (91) by passing to the limit in (138). Remark 5. When proving the estimates in Corollary2, Proposition8 and Propo- sition9 we only used the fact that un solves (92) with respect to the stochastic basis n (Ω, F, (Ft)t≥0, P, W, π). Since ue solves (138) with respect to the new stochastic basis (Ωe, Fe, (Fet)t≥0, Pe, W,f πe), the estimates (99), (100), (101), (102) ,(103) and (127) n also hold for the sequence (ue )n∈Λ on the probability space (Ωe, Fe, Pe). In particu- n 2 ∞ lar, the sequence (ue )n∈Λ is bounded, independently of n, in L (Ω;e L ([0,T ]; V )), L2(Ω;e L2([0,T ]; D(A))) and in L2(Ω;e W α,2([0,T ]; H)) for every α ∈ (0, 1/2). We now recall a suﬃcient condition for Lp convergence that will be used several times in this section. This is a version of Vitali’s convergence theorem.

Lemma 7.1. Let (Ωe, Fe, Pe) be a probability space, let X be a Banach space and let p p ∈ [1, ∞). Let f, f1, f2,... ∈ L (Ωe, Fe, Pe; X) and suppose that

i) |fn − f|X → 0 in probability as n → ∞ and q ii) sup Ee|fn|X < ∞ for some q ∈ (p, ∞). n≥1 p Then fn → f in the space L (Ωe, Fe, Pe; X). Lemma 7.1 is a simple measure theory exercise. The exponents will be p = 1 and q = 2 in our applications below. In addition, we will often have fn → f in X, Pe-a.s., which implies convergence in probability. Below we gather convergence results for each term that appears in the equation (138). We begin with a list of deterministic convergence results that imply that the candidate solution ue takes values in the space D(A). 8 4 2 Lemma 7.2. Suppose that Eku0k < ∞. Then ue ∈ L (Ω;e L ([0,T ]; D(A))) ∩ L4(Ω;e L∞([0,T ]; V )), Z T 2 2 Ee kue(s)k |Aue(s)| ds < ∞ (140) 0 and the following statements hold: n 1 1 i) ue → ue in L (Ω;e L ([0,T ]; V )) as n → ∞ along Λ. n 2 2 ii) ue * ue weakly in the space L (Ω;e L ([0,T ]; V )) as n → ∞ along Λ. n 4 2 iii) ue * ue weakly in the space L (Ω;e L ([0,T ]; D(A))) as n → ∞ along Λ. n ∗ 4 ∞ iv) ue → ue weak in the space L (Ω;e L ([0,T ]; V )) as n → ∞ along Λ. 5688 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

n 2 Proof.i ) From Proposition 14 we have ue → ue, Pe-a.s., in L ([0,T ]; V ) as n → ∞, n ∈ Λ. This convergence also holds in L1([0,T ]; V ), Pe-a.s. In order to show that this convergence holds in the space L1(Ω;e L1([0,T ]; V )) it is n suﬃcient, by Lemma 7.1, to observe that the sequence (ue )n∈Λ is bounded 2 1 n in L (Ω;e L ([0,T ]; V )). This follows from (100) which says that (ue )n∈Λ is 2 2 n bounded in L (Ω;e L ([0,T ]; V )). We conclude that (ue )n∈Λ converges to ue in the space L1(Ω;e L1([0,T ]; V )) as n → ∞, n ∈ Λ. ii) Fix a vector φ ∈ V and a set A ∈ Fe ⊗ B([0,T ]). Using the Cauchy-Schwarz inequality and part i) we see that Z T n Ee |((ue (s) − ue(s), φχA))| ds → 0 as n → ∞, n ∈ Λ. 0 By linearity it follows that Z T Z T n Ee ((ue (s), ψ(s))) ds −→ Ee ((ue(s), ψ(s))) ds as n → ∞, n ∈ Λ 0 0 for every measurable, V -valued simple function ψ on Ωe ×[0,T ]. Such functions 2 2 n are dense in the space L (Ω;e L ([0,T ]; V )), so it follows that ue → ue weakly in L2(Ω;e L2([0,T ]; V )) as n → ∞, n ∈ Λ. n 4 2 iii) The sequence (ue )n∈Λ is bounded in the space L (Ω;e L ([0,T ]; D(A))) by in- n equality (103). By Alaoglu’s theorem, every subsequence of (ue )n∈Λ has a further subsequence that converges weakly in L4(Ω;e L2([0,T ]; D(A))). We also have weak convergence to the same limit in the space L2(Ω;e L2([0,T ]; V )). But n 2 2 ue * ue weakly in L (Ω;e L ([0,T ]; V )) as n → ∞ along Λ by part ii), so it n 4 2 follows that ue * ue weakly in L (Ω;e L ([0,T ]; D(A))) as n → ∞ along Λ. n 4 ∞ iv) The sequence (ue )n∈Λ is bounded in the space L (Ω;e L ([0,T ]; V )) by inequal- n ity (101). By Alaoglu’s theorem, every subsequence of (ue )n∈Λ has a further subsequence that converges weak∗ in L4(Ω;e L∞([0,T ]; V )). We also have weak 2 2 n convergence to the same limit in the space L (Ω;e L ([0,T ]; V )). But ue * ue weakly in L2(Ω;e L2([0,T ]; V )) as n → ∞ along Λ by part ii), so it follows that n ∗ 4 ∞ ue → ue weak in L (Ω;e L ([0,T ]; V )) as n → ∞ along Λ. Finally, we have already observed in the proof of Proposition8 that the inequality 4 2 4 ∞ (140) follows from the fact that ue ∈ L (Ω;e L ([0,T ]; D(A))) ∩ L (Ω;e L ([0,T ]; V )).

In order to pass to the limit in (138) we will ﬁrst list convergence results that are required for analyzing the stochastic integral terms on the right-hand side of (138). The program for establishing these convergence results is essentially identical to the program in [9]. We will not go through the proofs here. Instead, we will state the relevant results as they appear in the present context.

n 2 Lemma 7.3. PnG(ue ) → G(ue) Pe-a.s., in the space L ([0,T ]; L2(U0,V )) as n → ∞ along Λ.

R t n Corollary 3. The processes ( PnG(u (s−)) dWf(s))t∈[0,T ] converge in prob- 0 e n∈Λ ability as L2([0,T ]; V )-valued random variables to (R t G(u(s−)) dW (s)) . 0 e f t∈[0,T ] Corollary3 follows from Lemma 7.3 by applying Proposition 4.16 in [11]. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5689

R t n Corollary 4. The processes ( PnG(u (s−)) dWf(s))t∈[0,T ] converge in the 0 e n∈Λ space L1(Ω; L1([0,T ]; V )) to (R t G(u(s−)) dW (s)) . e 0 e f t∈[0,T ] Corollary4 follows from Corollary3 by applying Lemma 7.1. In order to apply R t n this lemma, we are also required to verify that ( PnG(u (s)) dWf(s))t∈[0,T ] 0 e n∈Λ is a sequence bounded independently of n in the space L2(Ω;e L1([0,T ]; V )), but this follows easily from the growth condition (79) and inequality (101). Analogues of the three results above hold for the stochastic integrals with respect to the compensated Poisson random measure πeb in (138). In order to state these results succinctly we will write the coeﬃcient of dπeb in (138) as Ken(s, ξ) := n PnK(ue (s−), ξ) and the coeﬃcient of dπeb in (91) as Ke(s, ξ) := K(ue(s−), ξ). 2 Lemma 7.4. Ken → Ke as n → ∞ along Λ, Pe-a.s., in the space L ([0,T ]×E0, dt⊗ dν; V ). The following consequences of Lemma 7.4 can be established using similar arguments to the Wiener noise case above.

R R n Corollary 5. The processes ( PnK(u (s−), ξ) dπb(s, ξ))t∈[0,T ] con- (0,t] E0 e e n∈Λ verge in probability, as random variables with values in L2([0,T ]; V ), to the process R R ( K(u(s−), ξ) dπb(s, ξ))t∈[0,T ]. (0,t] E0 e e

R R n Corollary 6. The processes ( PnK(u (s−), ξ) dπb(s, ξ))t∈[0,T ] con- (0,t] E0 e e n∈Λ verge in the space 1 1 R R L (Ω; L ([0,T ]; V )) to the process ( K(u(s−), ξ) dπb(s, ξ))t∈[0,T ]. e (0,t] E0 e e 7.2. Passage to the limit. By combining the convergence results above we obtain the main result of this section, that ue satisfies (91) and is a martingale solution to the truncated equation (84). 8 Theorem 7.5. Suppose that Eku0k < ∞. Then ue is a global martingale solution to (84) with respect to the stochastic basis (Ωe, Fe, (Fet)t≥0, Pe, W,f πe) with initial condition ue0, where ue0 has the same law as that of u0. That is, ue solves equation (91). 2 2 2 ∞ Proof. By Lemma 7.2, ue ∈ L (Ω;e L ([0,T ]; D(A))) ∩ L (Ω;e L ([0,T ]; V )). The fact that ue ∈ D([0,T ]; H), Pe-a.s., is part of the conclusion of Proposition 14. We must show that equation (91) holds with ue(0) = ue0. Fix a vector φ ∈ D(A) and a set n 1 1 Γ ∈ Fe ⊗ B([0,T ]). Since ue → ue in the space L (Ω;e L ([0,T ]; V )) as n → ∞ in the set Λ by Lemma 7.2, we find that Z T n Ee χΓ(φ, ue (t) − ue(t)) dt → 0, 0 as n → ∞ along Λ. By Corollary4 we have Z T E χ (φ, R t[P G(un(s−)) − G(u(s−))] dW (s)) dt → 0, e Γ 0 n e e f 0 as n → ∞ along Λ. By Corollary6 we have Z T R R n Ee χΓ(φ, [PnK(u (s−), ξ) − K(u(s−), ξ)] dπb(s, ξ)) dt → 0, (0,t] E0 e e e 0 5690 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM as n → ∞ along Λ. We would like to prove a similar convergence result for the remaining terms on the left-hand side of equation (138). For the linear term we have Z T Z t Z T Z t n n Ee (Aue (s) − Aue(s), φχΓ) ds dt = Ee χΓ ((ue (s) − ue(s), φ)) ds dt 0 0 0 0 Z T n ≤ T kφkEe kue (s) − ue(s)k ds. (141) 0 The right-hand side of (141) tends to 0 as n → ∞ along Λ by part i) of Lemma 7.2. For the FR term in (138) we use the local Lipschitz condition (77b) to obtain Z T Z t Z T n n Ee (FR(ue (s)) − FR(ue(s)), φχΓ) ds dt ≤ TCR|φ|Ee kue (s) − ue(s)k ds. 0 0 0 (142) As above, the right-hand side of (142) tends to 0 as n → ∞ along Λ. It remains to consider the nonlinear B term on the right-hand side of (138). Let ue∗ be the solution to the random ODE (88) with initial condition ue0 and set ve(s) := ue(s) − ue∗(s). In preparation for the passage to the limit we use part i) of Lemma 7.2 to pass to a 0 n subsequence, indexed by some infinite set Λ ⊆ Λ, such that ue → ue in V , dPe ⊗ dt- 0 n a.e., as n → ∞ along Λ . Clearly, we have ve → ve in V , dPe ⊗ dt-a.e., as n → ∞ along Λ0 as well. By the triangle inequality we have Z t n n αn(t) := χΓ(φ, θ(kve (s)k)PnB(ue (s)) − θ(kve(s)k)B(ue(s))) ds 0 Z t n n ≤ θ(kve (s)k)χΓ(φ, PnB(ue (s)) − PnB(ue(s))) ds 0 Z t n + θ(kve (s)k)χΓ((I − Pn)φ, B(ue(s))) ds 0 Z t n + [θ(kve (s)k) − θ(kve(s)k)]χΓ(φ, PnB(ue(s))) ds 0 =: αn,1(t) + αn,2(t) + αn,3(t).

1 0 We claim that αn,j → 0 in L (Ωe × [0,T ]) as n → ∞, n ∈ Λ , for j = 1, 2, 3. We rewrite αn,1 using bilinearity of the nonlinear term, then use the cancellation property (73) and inequality (74) to obtain

αn,1(t) Z t n n n n = θ(kue (s)k)χΓ(φ, PnB(ue (s) − ue(s), ue(s)) + PnB(ue (s), ue (s) − ue(s))) ds 0 Z t n n n n ≤ θ(kve (s)k) (ue(s),B(ue (s) − ue(s),Pnφ)) + (ue (s) − ue(s),B(ue (s),Pnφ)) ds 0 Z T n n n ≤ C |APnφ|θ(kve (s)k)kue (s) − ue(s)k (kue(s)k + kue (s)k) ds 0 Z T n n n n ≤ C|Aφ| θ(kve (s)k)kue (s) − ue(s)k (kue(s)k + kve (s)k + kue∗ (s)k) ds. (143) 0

In the last step we used the fact that |APnφ| = |PnAφ| ≤ |Aφ|. We will estimate the right-hand side of (143) in two ways. First, using the cutoﬀ property (86) and REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5691 the Cauchy-Schwarz inequality we obtain Z T 1/2 n 2 2 n 2 αn,1(t) ≤ C|Aφ| kue − uekL2([0,T ];V ) (kue(s)k + κ + kue∗ (s)k ) ds 0 ! n n ≤ C|Aφ| kue − uekL2([0,T ];V ) · κ + sup [kue(s)k + kue∗ (s)k] . s∈[0,T ]

Note that sups∈[0,T ] kue(s)k is ﬁnite, Pe-a.s., by Lemma 7.2 and that n sup sup kue∗ (s)k < ∞, Pe-a.s., n≥1 s∈[0,T ]

n by inequality (89). By Proposition 14 we have kue − uekL2([0,T ];V ) → 0, Pe-a.s., as n → ∞ along Λ. Therefore, αn,1(t) → 0 as n → ∞ along Λ, Pe-a.s., for all t ∈ [0,T ]. Second, we use the triangle inequality and the cutoﬀ property (86) to estimate the right-hand side of (143) as Z T 2 2 n 2 αn,1(t) ≤ C|Aφ| kue(s)k + κ + kue∗ (s)k ds 0 ! 2 2 n 2 ≤ TC|Aφ| κ + sup [kue(s)k + kue∗ (s)k ] s∈[0,T ] ! 2 2 2 ≤ TC|Aφ| κ + sup [kue(s)k ] + kue0k . (144) s∈[0,T ] By Lemma 7.2, the right-hand side of inequality (144) (which does not depend on t) belongs to L1(Ωe × [0,T ]). Therefore, the dominated convergence theorem implies 1 that αn,1 → 0 in L (Ωe × [0,T ]) as n → ∞ along Λ. We estimate αn,2 using the cancellation property (73) and inequality (74) as above and obtain Z t n αn,2(t) = θ(kve (s)k)χΓ(ue(s),B(ue(s), (I − Pn)φ)) ds 0 Z T 2 ≤ C |A(I − Pn)φ| · kue(s)k ds 0 2 ≤ CT |(I − Pn)Aφ| · sup kue(s)k . (145) s∈[0,T ] The supremum on the right-hand side of (145) is ﬁnite Pe-a.s. by Lemma 7.2. Since |(I − Pn)Aφ| → 0, the right-hand side tends to 0, dPe ⊗ dt-a.e. Furthermore, since |(I − Pn)Aφ| ≤ |Aφ| we see that the right hand side of (145) is dominated by a function in L1(Ωe × [0,T ]). It follows from the dominated convergence theorem 1 that αn,2 → 0 in L (Ωe × [0,T ]) as n → ∞ along Λ. We estimate αn,3 using the cancellation property (73) and inequality (74) as above and obtain Z t n αn,3(t) = [θ(kve (s)k) − θ(kve(s)k)]χΓ(ue(s),B(ue(s),Pnφ)) ds 0 Z T n 2 ≤ C|APnφ| |θ(kve (s)k) − θ(kve(s)k)| · kue(s)k ds (146) 0 The dominated convergence theorem shows that the right hand side tends to 0 as n → ∞ along Λ0. Indeed, the integrand on the right-hand side of (146) is dominated 2 by the integrable function 2C|Aφ|kue(s)k and tends to 0, dPe ⊗ dt-a.e., as n → ∞ 5692 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

0 n 0 along Λ because ve → ve in V , dPe ⊗ dt-a.e., as n → ∞ along Λ and because the 1 cutoff function θ is continuous. This shows that αn → 0 in L (Ωe × [0,T ]) as n → ∞ along Λ0. We are now ready to collect results and finish the proof. 0 When we apply χΓ(φ, ·) to both sides of equation (138) and let n → ∞ along Λ in the space L1(Ωe ⊗ [0,T ]), we find that Z t Z t χΓ(φ, ue(t)) + χΓ(Aue(s), φ) ds + θ(kve(s)k)χΓ(φ, B(ue(s))) ds 0 0 Z t + χΓ(FR(ue(s)), φ) ds 0 Z t = χΓ(φ, ue0) + χΓ(φ, G(ue(s−)) dWf(s)) 0 Z Z + χΓ(φ, K(ue(s−), ξ)) dπeb(s, ξ), (0,t] E0 dPe ⊗ dt-a.e. Since D(A) is separable and dense in H, it follows that the equality above holds dPe ⊗ dt-a.e. for all φ ∈ H. This means that Z t Z t Z t χΓue(t) + χΓAue(s) ds + θ(kve(s)k)χΓB(ue(s)) ds + χΓFR(ue(s)) ds 0 0 0 Z t Z Z = χΓue0 + χΓG(ue(s−)) dWf(s) + χΓK(ue(s−), ξ) dπeb(s, ξ) (147) 0 (0,t] E0 in the space H, dPe ⊗ dt-a.e. Since ue is càdlàg in the H-norm, Pe-a.s., it follows (cf. Lemma 5.13 in [9]) that equation (91) holds for ue with initial condition ue0 with respect to the stochastic basis (Ωe, Fe, (Fet)t≥0, Pe, W,f πe). This shows that ue is a global martingale solution to (84).

Having established existence of a global martingale solution ue to the truncated equation (91) that is càdlàg in the H-norm, Pe-a.s., we now show that ue is actually càdlàg in the V -norm, Pe-a.s. We follow the argument given in Proposition 6.2 of [7], which extends the approach in Section 7.3 in [12] from the Wiener noise case to the Lévy noise case.

Proposition 15. Let (S,e ue) be a global martingale solution to the truncated equation (84). Then the sample paths of ue are càdlàg in the space V , Pe-a.s. Proof. Consider the linear, additive noise SDE: ( R dZ + AZ = G(u(t−)) dWf(t) + K(u(t−), ξ) dπb(t, ξ), e E0 e e (148) Z(0) = ue(0). Existence and uniqueness of a global solution Z to equation (148) can be established using a Galerkin method in a similar way to the approach given in Section5. We now sketch the argument and highlight its main conclusions. The Galerkin ∞ approximations, say (Zn)n=1, to (148) have càdlàg sample paths in the V -norm, Pe- a.s. In a similar way to Proposition8 one can show that the Galerkin approximations ∞ 4 ∞ 4 2 (Zn)n=1 are bounded in the space L (Ω;e L ([0,T ]; H)) ∩ L (Ω;e L ([0,T ]; D(A))). By applying the Itô formula with the function v 7→ kvk2 (as in Corollary1) to the diﬀerence Zn − Zm and making estimates as in Proposition8 one can show that the ∞ 2 ∞ sequence (Zn)n=1 is Cauchy in the space L (Ω;e L ([0,T ]; V )). Let Z denote the REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5693 limit of the sequence in this space. Then one can easily show that Z is the unique solution to (148). Furthermore, since Zn → Z in the V -norm, uniformly on [0,T ], Pe-a.s. along a subsequence it follows that Z is càdlàg in the V -norm, Pe-a.s. We will complete the proof by showing that u¯ := ue − Z is continuous in the V -norm, Pe-a.s. Observe that u¯ solves the random PDE ( d¯u + [Au¯ + θ(ku¯ + Zk)B(¯u + Z) + F (¯u + Z)] dt = 0 R (149) u¯(0) = 0.

Since u¯ ∈ L2(Ω;e L∞([0,T ]; V )) ∩ L2(Ω;e L2([0,T ]; D(A))), we see using (75) and (77a) that the terms Au¯, θ(ku¯ +Zk)B(¯u+Z) and FR(¯u+Z) all belong to the space 2 2 du¯ 2 2 L (Ω;e L ([0,T ]; H)), whence dt also belongs to L (Ω;e L ([0,T ]; H)). It follows from interpolation (see, e.g., Lemma 1.2 in Chapter 3 of [39]) that u¯ ∈ C([0,T ]; V ), Pe-a.s. We conclude that ue = Z +u ¯ is càdlàg in the V -norm, Pe-a.s. 8. Pathwise solution to the truncated equation. 8.1. Pathwise uniqueness. Deﬁnition 8.1. We say that pathwise uniqueness holds for the truncated equation (84) if for every pair of global martingale solutions u and v to (84) with respect to the same stochastic basis (Ω, F, (Ft)t≥0, P, W, π) one has

P[1{u(0)=v(0)}(u(t) − v(t)) = 0 ∀t ∈ [0,T ]] = 1. We will establish pathwise uniqueness for the truncated equation (84) in Propo- sition 16. In Theorem 8.2 we will employ a well-known argument that generalizes the Yamada-Watanabe theorem to obtain existence of a global pathwise solution to (84) from the existence of martingale solutions and pathwise uniqueness. In order to establish pathwise uniqueness for the truncated equation (84) we need to introduce stopping times to control certain terms that arise in the estimates. Let u and v be two local solutions to equation (84) up to a stopping time τ such that u, v ∈ L4(Ω; L∞([0,T ]; V )) ∩ L4(Ω; L2([0,T ]; D(A))). For each positive integer n, we deﬁne the random variable R t 2 2 2 2 τn := T ∧ inf{t ∈ [0, τ): 0 [ku(s)k · |Au(s)| + kv(s)k · |Av(s)| ] ds ≥ n}, (150) R t 2 2 2 2 Note that the integral φ(t) := 0 [ku(s)k · |Au(s)| + kv(s)k · |Av(s)| ] ds is ﬁnite, P-a.s., due to the fact that u, v ∈ L4(Ω; L∞([0,T ]; V )) ∩ L4(Ω; L2([0,T ]; D(A))). Since φ is adapted and continuous in t a.s. it follows that τn is an Ft-stopping time (see, e.g., Proposition 4.5 in Chapter I of [35]). We are now ready to establish local pathwise uniqueness of solutions to the truncated equation (84) (cf. Proposition 5.1 in [12]). Proposition 16. Pathwise uniqueness holds for the truncated equation (84). More- over, if u and u0 are local pathwise solutions to (84) up to the same stopping time τ relative to the same stochastic basis (Ω,F,(Ft)t≥0, P, W, π) with respective initial 0 conditions u0 and u0, then we have 0 P[1 0 (u(t) − u (t)) = 0 ∀t ∈ [0, τ)] = 1. {u0=u0} 0 0 Proof. Let w(t) := 1Ω0 (u(t) − u (t)) and Ω0 := {u0 = u0}. We will show that ! E sup kw(t)k2 = 0. (151) t∈[0,τ) 5694 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

0 Let u∗ and u∗ solve the random ODE (88) with respective initial conditions u0 and 0 0 0 0 u0. Deﬁne v := u − u∗ and v := u − u∗. We have the following SDE for w:

0 0 0 dw + [Aw + θ(kvk)B(u) − θ(kv k)B(u ) + FR(u) − FR(u )]1Ω0 dt 0 = 1Ω0 (G(u(t−)) − G(u (t−))) dW (t) Z + 1 (K(u(t−), ξ) − K(u0(t−), ξ)) dπ(t, ξ), (152) Ω0 b E0 w(0) = 0.

We would like to apply the Itô formula (Theorem 2.19) to w in (152) using the function ψ : D(A) → R deﬁned by ψ(x) := |Ax|2. Even though some of the terms in equation (152) are only H-valued a priori and not necessarily D(A)-valued, the terms in the right-hand side of the resulting Itô formula (i.e., equation (26)) are still well-deﬁned and the conclusion of the Itô formula is still valid (see, e.g., Proposition 2 in Chapter VI of [27]). In the present case we obtain

Z t kw(t)k2 + 2 |Aw(s)|2 ds 0 Z t = 2 (θ(kv0(s)k)B(u0(s)) − θ(kv(s)k)B(u(s)), Aw(s)) ds 0 Z t 0 + 2 (FR(u (s)) − FR(u(s)), Aw(s)) ds 0 Z t + 2 ((w(s), [G(u(s−)) − G(u0(s−))] dW (s))) 0 Z t + kG(u(s)) − G(u0(s−))k2 ds L2(U0,V ) 0 Z Z 0 + 2 ((w(s−),K(u(s−), ξ) − K(u (s−), ξ))) dπb(s, ξ) (0,t] E0 Z Z + kK(u(s−), ξ) − K(u0(s−), ξ)k2 dπ(s, ξ) (0,t] E0

=: I1(t) + I2(t) + I3(t) + I4(t) + I5(t) + I6(t), (153) a.s. for all t ∈ [0, τ). In order to prove (151) it is suﬃcient, by the monotone convergence theorem, to prove that ! E sup kw(t)k2 = 0, (154) t∈[0,τn∧τ] for each positive integer n. Following the proof of Proposition 5.1 in [12], we will establish (154) using the stochastic Gronwall Lemma established in [18]. In preparation, let τa < τb be Ft-stopping times such that τb ≤ τn ∧ τ, P-a.s. Using the Lipschitz condition (79), we easily obtain

" # Z τb 2 E sup [|I4(s)| + |I6(s)|] ≤ CE kw(s)k ds. (155) s∈[τa,τb] τa REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5695

Using the BDG inequalities (18) and (24), Young’s inequality and the Lipschitz condition (79) we obtain " # " # Z τb 1 2 2 E sup [|I3(s)| + |I5(s)|] ≤ E sup kw(s)k + E kw(s)k ds. (156) s∈[τa,τb] 2 s∈[τa,τb] τa Using the Cauchy-Schwarz inequality, (77b) and Young’s inequality we ﬁnd that there exists a constant C = C(R) > 0 such that " # Z τb Z τb 1 2 2 E sup |I2(s)| ≤ E |Aw(s)| ds + CE kw(s)k ds. (157) s∈[τa,τb] 2 τa τa

We estimate J1 using the fact that the cutoﬀ function θ is Lipschitz and inequality (75) to obtain

Z τb J1 ≤ CE kw(s)k · ku(s)k · |Au(s)| · |Aw(s)| ds. τa By applying Young’s inequality we find that there exists a constant C > 0 such that Z τb Z τb 1 2 2 2 2 J1 ≤ E |Aw(s)| ds + C kw(s)k ku(s)k |Au(s)| ds. (159) 2 τa τa To estimate J2 we use the bilinearity of B, inequality (75) and Young’s inequality to obtain Z τb 0 J2 ≤ E [|(B(w(s), u(s)), Aw(s))| + |(B(u (s), w(s)), Aw(s))|] ds τa Z τb h i ≤ CE |Aw(s)|3/2kw(s)k1/2 · ku(s)k1/2|Au(s)|1/2 + ku0(s)k1/2|Au0(s)|1/2 ds. τa 1 Z τb Z τb ≤ E |Aw(s)|2 ds + C kw(s)k2 ku(s)k2|Au(s)|2 + ku0(s)k2|Au0(s)|2 ds. 2 τa τa (160) We are now ready to collect the estimates above and complete the proof. Applying inequalities (156), (157), (159) and (160) in (153) and using (155) we find that h i E sup kw(s)k2 s∈[τa,τb] (161) Z τb ≤ CE kw(s)k2 1 + ku(s)k2|Au(s)|2 + ku0(s)k2|Au0(s)|2 ds. τa In order to apply the stochastic Gronwall Lemma of [18], we observe that the process φ(s) := 1 + ku(s)k2|Au(s)|2 + ku0(s)k2|Au0(s)|2 satisfies

Z τn∧τ φ(s) ds ≤ n, P-a.s.. 0 5696 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

As a consequence, we have ! Z τn∧τ h Z τn∧τ i E kw(s)k2φ(s) ds ≤ E sup kw(s)k2 φ(s) ds 0 s∈[0,τn∧τ] 0 ! ≤ nE sup kw(s)k2 s∈[0,τn∧τ] < ∞, because w ∈ L2(Ω; L∞([0,T ]; V )) . We may now apply the stochastic Gronwall inequality from [18] to deduce (154), which completes the proof. We have established existence of global martingale solutions to the truncated equation (84) in Theorem 7.5 and we have established pathwise uniqueness in Propo- sition 16. One can now apply a well-known generalization of the Yamada-Watanabe theorem in [41], based on a result of Gyöngy and Krylov in [19], to deduce the existence of a unique global pathwise solution to (84) (see, e.g., page 1135 in [12]). We state the conclusion below.

Theorem 8.2. Let (Ω, F, (Ft)t≥0, P, W, π) be a stochastic basis. Fix a value of the 8 cutoﬀ parameter R > 0 and let u0 ∈ L (Ω, F0, P; V ). Then there exists a unique global pathwise solution u to the truncated equation (84). Furthermore, we have u ∈ L4(Ω; L∞([0,T ]; V ))∩L4(Ω; L2([0,T ]; D(A))), u is càdlàg in the V -norm P-a.s. and n ∞ the solutions (u )n=1 to (92) converge to u in probability as Xu = D([0,T ]; H) ∩ L2([0,T ]; V )-valued random variables.

9. Maximal local solution to the main SPDE with bounded jump noise. The goal of this section is to remove the truncation from the drift term FR in equation (84) thus obtaining a local pathwise solution to (83), the main SPDE with bounded jump noise. Furthermore, we will relax the moment requirement (90), 8 that u0 ∈ L (Ω; V ), and instead take any given F0-measurable, V -valued random variable u0 as the initial condition. At this point we have found global pathwise solutions to the truncated equation (84) for each ﬁxed cutoﬀ parameter R > 0. In this section we will let R vary. Hereafter we will make the dependence on R explicit and refer to the solution of equation (84) as uR.

9.1. Removing the truncation. Let (Ω, F, (Ft)t≥0, P, W, π) be a stochastic basis and let u0 :Ω → V be F0-measurable. We will construct a maximal local solution to (162) up to a strictly positive stopping time a.s. For each R > 0 there exists a global pathwise solution uR to equation (84) with initial condition u01{ku0k 0 : kuR(t)k > R}. (163)

Equation (162) is just equation (84) with FR replaced by F . Note that since uR is adapted and càdlàg in the V -norm a.s. it follows from standard results that σR is a stopping time (see, e.g., Proposition 4.8 in Chapter I of [35]). The construction of a REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5697 maximal local solution to equation (162) from the local solutions (uR, σR) is similar to the construction of maximal local solutions to equation (58) in Subsection 3.2. For instance, an application of Proposition 16 shows that (cf. (64))

uR1 (t) = uR2 (t) for all t ∈ [0, σR1 ∧ σR2 ), P-a.s. on the event {ku0k ≤ R1}. (164) In a similar way to the proof of Lemma 3.4 we deduce the following from (164):

Lemma 9.1. In the setting above, in which u0 :Ω → V is F0-measurable, we have, for all R1 < R2

i) σR2 > 0, P-a.s. on {ku0k < R1},

ii) σR1 ≤ σR2 , P-a.s. on {ku0k < R1}, and iii) uR1 (σR1 ) = uR2 (σR1 ), P-a.s. on {ku0k < R1}. The existence time of the maximal local solution to equation (162) is

σ := lim sup σR = lim σR. (165) R→∞ R→∞ The maximal local solution to equation (162) is the process u:Ω × [0,T ] → D(A) deﬁned by (cf. (68)) ( lim uR(t) if t < σ u(t) := R→∞ (166) 0 otherwise.

The limit in (166) exists P-a.s. because the sequence uR(t) stabilizes for all R > 0 such that t < σR. The next result can be proved in the same way as Proposition6.

∞ Proposition 17. In the setting above with u0 ∈ L (Ω, F0, P; V ), the following statements hold true: i) P[σ = 0] = 0. S ii) P R>0{σR = σ, σ < T } = 0. iii) sup ku(t)k = ∞, P-a.s. on the event {σ < T }. t∈[0,σ) h 2i iv) If E supt∈[0,σ) ku(t)k < ∞, then P[σ = T ] = 1.

We now return to equation (83) by removing the cutoﬀ function θ from equation (162). Let u0 :Ω → V be F0-measurable and let (u, σ) be the maximal local pathwise solution to (162) with initial condition u0 as in (166). It is clear that u is a local solution to equation (83) up to the stopping time

τ := σ ∧ inf{t > 0 : ku(t) − u∗(t)k > κ/2}. (167)

Since the diﬀerence u−u∗ starts at 0 and is right-continuous in the V -norm at time t = 0, P-a.s., it follows that τ > 0, P-a.s. We record below the properties of the local pathwise solution that we have just established above.

Theorem 9.2. Let (Ω, F, (Ft)t≥0, P, W, π) be a stochastic basis and let u0 :Ω → V be an arbitrary F0-measurable random variable. Then there exists an Ft-adapted, D(A)-valued process (u(t))t∈[0,T ] and an Ft-stopping time τ such that τ > 0 a.s., u is càdlàg in the V -norm on [0, τ) a.s., the pair (u, τ) is a local solution to equation (83), i.e., (u, τ) satisﬁes the integral equation (81) with K = 0. 5698 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

9.2. Examples of global existence. In this section we give some examples of speciﬁc SPDEs for which the local solution to equation (1) is actually global, i.e., τ = T a.s. In view of the piecing-out argument, which appears in the next section, it is suﬃcient to establish global existence in the case of bounded jump noise, i.e., in the setting of equation (83). We continue to assume that G and K satisfy conditions (78) and (79).

Example 2. Consider (83) with B ≡ 0 and a globally Lipschitz function F : V → H. In this case, F satisﬁes conditions (77a) and (77b) with a constant that does not depend on R, so there is no need to consider the truncated equation (84). The exact same reasoning that was applied to equation (84) in Sections5,6,7 and 8 can be applied directly to equation (83). When the initial condition satisﬁes 2 u0 ∈ L (Ω, F0, P; V ), we obtain a unique global pathwise solution to equation (83) that possesses all of the properties listed in Theorem 8.2. In order to solve the equation starting at an arbitrary F0-measurable initial condition u0 :Ω → V , we can again consider solutions with u01{ku0k

Example 3 (Chafee-Infante Equation). This example appears in [13]. Consider the SPDE

 3 R ∂tu − ∆u + λ(u − u) = G(u) dW + K(u, ξ) dπ in (0,T ) × O  E0 b u = 0 on (0,T ) × ∂O (168)  u(0) = u0 on {0} × O, where O is an open, bounded subset of Rd, d ≤ 3, with smooth boundary ∂O. To put equation (168) in the form of the abstract SPDE (83) we take H := L2(O) and 1 V := H0 (O). For the linear operator A we take the negative Laplacian A := −∆. There is no bilinear term, i.e., we set B = 0. The nonlinearity F : V → H is given by F (u) := λ(u3 − u), for all u ∈ V, (169) where λ > 0. Note that by Sobolev embedding we have V ⊆ L6(O) when d ≤ 3, so F is well-deﬁned on all of V . It is clear that F satisﬁes the local growth and Lipschitz conditions (77a) and (77b). Let (u, τ) be the maximal local pathwise solution to equation (168) on the time interval [0,T ] (which exists by Theorem 9.2). We will show that " # E sup ku(s)k2 < ∞, (170) s∈[0,τ) which then implies that τ = T a.s. by Proposition 17. We apply the Itô formula to u using the function ψ(v) := |Av|2 (see Proposition 2 in Chapter VI of [27]) and obtain Z t kun(t)k2 + 2 [|Aun(s)|2 + (Au(s),F (u(s)))] ds 0 Z t 2 n = ku0k + 2 ((u(s),G(u (s−)) dW (s))) 0 REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5699 Z Z n + 2 ((u(s),K(u (s−), ξ))) dπb(s, ξ) (171) (0,t] E0 Z t + kG(u(s−))k2 L2(U0,V ) 0 Z Z + kK(u(s−), ξ)k2 dπ(s, ξ), (0,t] E0 R t P-a.s. for all t ∈ [0, τ). We focus on the term 0 (Au(s),F (u(s))) ds. All of the remaining terms will be handled in an identical way to the proof of Proposition8. Using integration by parts, we see that

(Av, F (v)) = λ(v3, −∆v) − λ(v, −∆v) Z = 3λ |v|2|∇v|2 dx − λkvk2. O When we apply the inequality above to (171), we drop the ﬁrst term, which is positive, and move the second term to the right-hand side of (171). Then, reasoning as in the proof of Proposition8, we obtain

" # Z t∧τ Z t " # E sup ku(s)k2 + E |Au(s)|2 ds ≤ C + C E sup ku(r)k2 ds. s∈[0,t∧τ) 0 0 r∈[0,s∧τ) (172) We obtain (170) by applying Gronwall’s inequality to the estimate above and conclude that P[τ < T ] = 0. So, the maximal pathwise solution u to the Chafee-Infante equation (168) is a global solution.

10. Local pathwise solutions to the main SPDE. The goal of this section is to prove existence and uniqueness of a local pathwise solution to equation (1). In Theorem 9.2 above we showed that equation (1) possesses a local pathwise solution (u, τ), when K ≡ 0, for every F0-measurable initial condition u0 :Ω → V . We will use the piecing-out argument introduced by [20] to construct a local pathwise solution to equation (1) in the case where K is not identically zero. A similar construction in the case of globally defined, finite dimensional SDEs is given in [2] (where it is referred to as the interlacing argument) and in Theorem 9.1 of [21]. We give a sketch of the piecing-out argument now in order to identify some of the underlying probabilistic notions that are at work. Recall that we assume that π is the Poisson random measure corresponding to a stationary Ft-Poisson point process Ξ on a measurable space (E, E). The intensity measure of π (which is a measure on (0, ∞) × E) is dt ⊗ dν and we assume that ν is a σ-finite measure on (E, E) such that ν(E \ E0) < ∞. The point process Ξ is a random partial function Ξ: (0, ∞) *E whose domain, denoted DΞ, is a countable subset of (0, ∞) a.s. What makes Ξ a stationary Poisson point process is that for every measurable set Γ ∈ B((0, ∞)) ⊗ E the quantity π(Γ) = #{s ∈ DΞ :(s, Ξ(s)) ∈ Γ} is a Poisson R random variable with mean Γ dt dν. What makes Ξ an Ft-point process is that for every measurable set A ∈ E the Poisson process (π((0, t] × A))t≥0 is adapted to (Ft)t≥0 and for all t > s > 0 the random variable π((s, t] × A) is independent of Fs. The key idea behind the piecing-out argument is that the stochastic integral with respect to the random measure π is simply a sum of finitely many vectors in H a.s. 5700 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Indeed, we have Z Z X K(u(s−), ξ) dπ(s, ξ) = K(u(s−), Ξ(s))χE\E0 (Ξ(s)) (0,t] E\E0 s∈(0,t]∩DΞ by Theorem 2.16 and the number of terms in the sum is a Poisson random variable with mean t · ν(E \ E0), which we assume is ﬁnite. Let us enumerate the elements of {s ∈ DΞ : Ξ(s) ∈ E \ E0} as σ1 < σ2 < σ3 < ··· . The method in the piecing-out argument is to construct a solution to (1) by solving equation (83) up to time σ1 ∧τ, where τ is the time of existence of the local solution to (83). If τ ≤ σ1, then the solution to (83) blows up at or before the jump at time σ1, so the construction stops. Otherwise, if σ1 < τ, then we construct a solution to (1) on the interval [σ1, σ2) by solving (83) with initial condition u(σ1−) + K(u(σ1−), Ξ(σ1)) and with noise restarted at time σ1. Repeating this procedure inductively yields a local solution to (1). Before we can give a detailed proof of the piecing-out argument we explain what is meant by “restarting” the noise at the jump times σ1 < σ2 < ··· . The strong Markov property, which we review below, says that if we look at a Lévy process L after a stopping time τ and translate it to start at zero, then we obtain a Lévy process with the same distribution as L. More precisely, Le(t) := L(t + τ) − L(τ) is a Lévy process with the same distribution as L. We use the strong Markov property to change variables in stochastic integrals on random time intervals in the piecing-out argument. We review the strong Markov property and prove the change of variables arguments in Subsection 10.1, then we apply these results during the piecing-out argument in Subsection 10.2. 10.1. Shifting by a stopping time. In this section we recall the strong Markov property and establish technical lemmas related to random changes of variables in stochastic integrals. We begin by recalling the strong Markov property and some of its corollaries. In order to state the strong Markov property, we need to recall the σ-ﬁeld associated to a stopping time.

Definition 10.1. Let (Ω, F, (Ft)t≥0, P) be a filtered probability space. We define W S F∞ := t>0 Ft to be the σ-field generated by the sets t>0 Ft. For every Ft- stopping time τ we define

Fτ := {A ∈ F∞ : A ∩ {τ ≤ t} ∈ Ft for every t ≥ 0}. (173)

Since {τ ≤ t} ∈ Ft we see that Fτ is a σ-ﬁeld.

Lemma 10.2. Let (Ω, F, (Ft)t≥0, P) be a ﬁltered probability space satisfying the usual conditions, let L be a U-valued Ft-Lévy process and τ be an Ft-stopping time. Then the following statements hold: i) L is a Feller process. ii) L satisﬁes the strong Markov property, i.e., for every nonnegative (or bounded, real-valued), F∞-measurable random variable Z we have

E[Z ◦ θτ | Fτ ] = EL(τ)[Z] (174) P-a.s., where θ is the left shift on the canonical path space. iii) τ + t is an Ft-stopping time for each t ≥ 0. iv) Le(t) := L(t+τ)−L(τ) is a U-valued Lévy process that is adapted to the ﬁltration (Fτ+t)t≥0 and has the same distribution as L. v) Le(s + t) − Le(s) is independent of Fτ+s for all s, t ≥ 0. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5701

vi) Le is independent of the σ-field Fτ . Remark 6. The name “strong Markov property” refers to the property in (174). We give a brief explanation of the strong Markov property next. In order to do this we must first recall some concepts from the theory of Markov processes. Since L is a Markov process it has a transition semigroup (which is given explicitly on page 39 in [32]). The transition semigroup does not determine the distribution of Markov process completely, the distribution of the initial value is also required to specify the distribution of a unique process; this is Theorem 1.5 in [35], page 82. For Lévy processes we impose the condition that L(0) ≡ 0. One typically thinks of a Markov process on U as the family of U-valued processes obtained from a fixed transition semigroup by using each Borel probability measure on U as the initial measure. The same letter is typically used for the process while the initial measure is specified in the notation by a subscript. For example, Pµ is the law of the Markov process with initial measure µ and Eµ is the associated expectation. For the Dirac measures

µ = δx, x ∈ U, it is common to write Px, Ex instead of Pδx , Eδx . We adopt this convention in the statement of (174), using L to denote the process, even though the distribution of L under Pµ is only a Lévy process when µ = δ0. For this reason we have P0 = P and E0 = E, so we omit the subscript 0 from the notation. Finally, on the left-hand side of (174) we have a conditional expectation with respect to the probability measure P and on the right-hand side we have the random variable ω 7→ EL(τ(ω))[Z], i.e., the expectation of Z using the vector L(τ(ω)) as the initial value of the process. As we will see in iv), the strong Markov property can be used to justify the intuition (based on the stationary increments property) that a Lévy process “restarts” with the same distribution at a stopping time after translating its initial value to 0. Property vi) justiﬁes the intuition (based on the independent increments property) that the process Le restarted at τ is independent of events that occur before time τ. Proof of Lemma 10.2.i ) Every Lévy process is a Markov process and the transition semigroup of a Lévy process is given by a convolution semigroup of measures ([32], page 39). Using Proposition 2.4 in [35] it is easy to check that the transition function of a Lévy process satisﬁes the Feller property, which makes the Lévy process a Feller process. ii) All Feller processes satisfy the strong Markov property. This is Theorem 3.1 in [35]. iii) This is straightforward; it is also Lemma 3.2 in [35]. iv) The random variable L(t + τ) is Fτ+t-measurable for every t ≥ 0 by Propo- sitions 4.8 and 4.9 in [35]. Since τ ≤ τ + t for t ≥ 0 we have Fτ ⊆ Fτ+t by Exercise 4.16 in Chapter I of [35] and we see that Le(t) is Fτ+t-measurable. It follows from the strong Markov property that Le inherits stationary, independent increments and stochastic continuity from L. For instance, to show that the increments of Le are independent let 0 ≤ s1 < t1 ≤ s2 < t2 ≤ · · · ≤ sm < tm, let Γ1,..., Γm ∈ B(U) and take Z in (174) to be the indicator function of the Tm event j=1{L(tj) − L(sj) ∈ Γj}. The strong Markov property says that m m h \ i h \ i P {Le(tj) − Le(sj) ∈ Γj} = P {L(tj + τ) − L(sj + τ) ∈ Γj} j=1 j=1

= E[Z ◦ θτ ] 5702 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

= E EL(τ)[Z] m h h \ ii = E PL(τ) {L(tj) − L(sj) ∈ Γj} j=1 m h Y i = E PL(τ)[L(tj) − L(sj) ∈ Γj] . (175) j=1 In the last step we use the fact that the distribution of the increments of L does not depend on the initial distribution, in particular, under PL(τ(ω)) they are independent for all ω ∈ Ω. Moreover, since the integrand in the last line does not depend on ω, we can interchange the expectation and the product. Therefore, m m h \ i Y P {Le(tj) − Le(sj) ∈ Γj} = E PL(τ)[L(tj) − L(sj) ∈ Γj] j=1 j=1 m (176) Y = P[Le(tj) − Le(sj) ∈ Γj]. j=1 The second step follows because for 1 ≤ j ≤ m the reasoning that led to (175) also applies to the indicator function of the event {L(tj) − L(sj) ∈ Γj} and yields the same result (without the intersection on the left-hand side and without the product on the right-hand side). The second equality in (176) shows that Le has independent increments. The other applications of the strong Markov property that we require are very similar to the one above. Since Le(0) = 0 we see that Le is a Lévy process. The strong Markov property implies that Le and L have the same ﬁnite dimensional marginal distributions. v) This property is inherited by Le from L by the strong Markov property. vi) This is Corollary 3.6 in [35]; it is another application of the strong Markov property. We now state the precise ways in which we will apply the strong Markov property. Let (Ω, F, (Ft)t≥0, P, W, π) be a stochastic basis and let τ be an Ft-stopping time. By Lemma 10.2, the process Wf(t) := W (t + τ) − W (τ) (177) is an Ft+τ -Wiener process with the same covariance operator as W . We also consider the shifted process Ξ(e t) := Ξ(t + τ) with domain D := {s > 0 : s + τ ∈ DΞ}, P Ξe and let π := s∈D δ . For each set Γ ∈ B((0, ∞)) ⊗ E, Lemma 10.2 implies e Ξe (s,Ξ(e s)) that (πet(Γ))t≥0 is an Ft+τ -Poisson process with the same distribution as (πt(Γ))t≥0. So πe is a Poisson random measure with the same distribution as π. This means that Ξe is a stationary Poisson point process on E. The overall conclusion is that the shifted stochastic basis (Ω, F, (Ft+τ )t≥0, P, W,f πe) has the same properties, in distribution, as the original stochastic basis (Ω, F, (Ft)t≥0, P, W, π). Furthermore, the shifted ﬁltration (Ft+τ )t≥0 continues to satisfy the usual conditions. Since F0 ⊆ Fτ we see that (Ft+τ )t≥0 is complete. The fact that (Ft+τ )t≥0 is right-continuous follows from Exercise 4.17 in Chapter I of [35]. With the strong Markov property in hand we are now able to state and prove a change of variables formulas for stochastic integrals with respect to Wiener processes. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5703

Lemma 10.3. Let (Ω, F, (Ft)t≥0, P) be a filtered probability space satisfying the usual conditions, let W be a U-valued Ft-Wiener process and let τ be a stopping time with τ ≤ T a.s. Consider the identically distributed Wiener process Wf(t) := W (t + τ) − W (τ) and let L2 (H) denote the space of integrands with respect to U0,T,τ 2 Wf, i.e., the processes in L (Ω × [0,T ]; L2(U0,H)) that are predictable with respect to the shifted filtration (F ) . For each Ψ ∈ L2 (H) the process (ω, s) 7→ t+τ t≥0 U0,T,τ Ψ (s − τ(ω))χ (s) belongs to L2 (H). This transformation is a bounded ω (τ(ω),T ] U0,T linear map from L2 (H) → L2 (H) and satisfies U0,T,τ U0,T Z t−τ Z t χ{τ

0 < s1 < s2 ≤ T , A ∈ Fs1+τ and Φ ∈ L(U, H). Then Ψ is a simple process that is predictable with respect to the shifted ﬁltration (Ft+τ )t≥0. Furthermore, every such simple process is a ﬁnite sum of processes of the form Ψ supported on disjoint time intervals. This remains true if we impose the additional condition that s2 − s1 ≤ min{tk − tk−1 : 2 ≤ k ≤ m}. We can write

Ψ(s − τ)χ(τ,T ](s) = χAχ(s1+τ,s2+τ](s)χ(τ,T ](s)Φ m X = χA∩{τ=tk}χ(s1+tk,s2+tk](s)χ(tk,T ](s)Φ k=1 m0 X = χA∩{τ=tk}χ(s1+tk,s2+tk](s)Φ, (179) k=1 0 where m ≤ m is maximal such that s2 + tm0 < T . The time intervals above are disjoint because s2 − s1 ≤ tk − tk−1 for 2 ≤ k ≤ m. The event A ∩ {τ = tk} is

Fs1+tk measurable because

A ∩ {τ = tk} = A ∩ {s1 + τ = s1 + tk} c = (A ∩ {s1 + τ ≤ s1 + tk}) ∩ {s1 + τ < s1 + tk} .

The event in parentheses belongs to Fs1+tk because A is Fs1+τ measurable, so does the event on the right because s1 + τk is an Ft-stopping time. Therefore, the right- hand side of (179) is a simple process that is predictable with respect to (Ft)t≥0. Consequently, for t ∈ [0,T ] we have Z t Ψ(s − τ)χ(τ,T ](s) dW (s) 0 m0 X = χA∩{τ=tk}Φ(W ((s2 + tk) ∧ t) − W ((s1 + tk) ∧ t)) k=1 m X = χA∩{τ=tk}Φ(W ((s2 + tk) ∧ t) − W ((s1 + tk) ∧ t)) k=1

= χAΦ(W ((s2 + τ) ∧ t) − W ((s1 + τ) ∧ t)) 5704 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

= χAΦ(W ((s2 ∧ (t − τ)) + τ) − W ((s1 ∧ (t − τ)) + τ))

= χAΦ(Wf(s2 ∧ (t − τ)) − Wf(s1 ∧ (t − τ))) Z t−τ = χ{τ m , so all terms with k > m0 are zero. Since both sides of (178) are linear in Ψ it follows that (178) holds for all simple processes in L2 (H). U0,T,τ Now we remove the restriction on τ. Let τ be any Ft-stopping time that is ∞ bounded above by T a.s. There exists a sequence of Ft-stopping times (τn)n=1, each bounded above by T and taking ﬁnitely many values a.s., such that τn ↓ τ. Let Ψ be a simple process that is predictable with respect to (Ft+τ )t≥0. Since

τ ≤ τn for each n we have Ft+τ ≤ Ft+τn by Exercise 4.16 in Chapter I of [35].

Therefore, Ψ is predictable with respect to (Ft+τn )t≥0. Since τn ↓ τ we see that

χ{τn

Z t−τn Z t−τ Ψ(s) dWf(s) → Ψ(s) dWf(s) 0 0 because the stochastic integral with respect to Wf is continuous a.s. On the other hand, since Ψ is left-continuous and τn ↓ τ we see that Ψ(s − τn)χ(τn,T ](s) → Ψ(s − τ)χ(τ,T ](s) dP ⊗ ds-a.e. Using the dominated convergence theorem we see that this convergence holds in the space L2 (H) (the integrands are all bounded U0,T by the same constant). Therefore, Ψ(s − τ)χ (s) belongs to the space L2 (H) (τ,T ] U0,T and for each t ∈ [0,T ] we have Z t Z t

Ψ(s − τn)χ(τn,T ](s) dW (s) → Ψ(s − τ)χ(τ,T ](s) dW (s) 0 0 in the space L2(Ω; H). So, we have established (178) for an arbitrary stopping time τ and simple processes Ψ that are predictable with respect to the shifted ﬁltration

(Ft+τ )t≥0. Finally, let Ψ ∈ L2 (H) and let (Ψ )∞ be a sequence of simple processes in U0,T,τ n n=1 L2 (H) that converge to Ψ in the space L2(Ω × [0,T ]; L (U ,H)). Since U0,T,τ 2 0 Z T Z T −τ E kΨ(s − τ) − Ψ (s − τ)k2 ds = E kΨ(s) − Ψ (s)k2 ds, n L2(U0,H) n L2(U0,H) τ 0 (180) we see that Ψ(s − τ)χ (s) belongs to the space L2 (H). Similarly, the rule (τ,T ] U0,T Ψ 7→ Ψ(s − τ)χ (s) deﬁnes a bounded mapping from L2 (H) → L2 (H) (τ,T ] U0,T,τ U0,T with norm at most 1. For each ﬁxed t ∈ [0,T ], we have Z t Z t Ψn(s − τ)χ(τ,T ](s) dW (s) → Ψ(s − τ)χ(τ,T ](s) dW (s) 0 0 in L2(Ω; H) by (180). By passing to a subsequence we see that the convergence above holds a.s. in H. Since (178) holds for each Ψn, we see that the almost sure R t−τ limit is χ{τ

Lemma 10.4. Let (Ω, F, (Ft)t≥0, P) be a filtered probability space satisfying the usual conditions. Let Ξ be a stationary Ft-Poisson point process on a measurable space (E, E), let π be its corresponding Poisson random measure and let dν ⊗ dt be the intensity measure of π. Let τ be an Ft-stopping time with τ ≤ T a.s. and q define Ξ(e t) := Ξ(t + τ) and πe as above. For q = 1, 2 let Fν,T,τ (H) denote the elements of Lq(Ω × [0,T ] × E; H) that are predictable with respect to the shifted 2 filtration (Ft+τ )t≥0. For each f ∈ Fν,T,τ (H) the function (ω, s, ξ) 7→ f(ω, s − 2 τ(ω), ξ)χ(τ(ω),T ](s) belongs to Fν,T (H). This defines a bounded linear map from 2 2 Fν,T,τ (H) → Fν,T (H) that satisfies Z Z Z Z χ{τ

2 in H a.s. for all f ∈ Fν,T,τ (H) and all t ∈ [0,T ]. Proof. The method is similar to the proof of Lemma 10.3 except that in place of simple processes we use simple functions ψ :Ω × [0,T ] × E → H of the form

m−1 X ψ(ω, s, ξ) = χAj (ω)χ(sj ,sj+1](s)χEj (ξ)xj (182) j=1

where Aj ∈ Fsj +τ , 0 < sj < sj+1 ≤ T , Ej ∈ E with ν(Ej) < ∞ and xj ∈ H, and the rectangles Aj × (sj, sj+1] × Ej are disjoint. If τ takes ﬁnitely many values a.s., then a similar argument to the one in Lemma 10.3 shows that ψ(s − τ, ξ)χ(τ,T ](s) is a simple function that is Ft-predictable. For general τ, if τn ↓ τ where τn ≤ T take

ﬁnitely many values a.s., then we see that ψ(s−τn, ξ)χ(τn,T ](s) → ψ(s−τ, ξ)χ(τ,T ](s) in H, dP⊗ds⊗dν-a.e. In order to establish this convergence result we use the fact that ψ(ω, ·, ξ) is left-continuous for all (ω, ξ) ∈ Ω × E. Simple functions of the form 2 2 ∞ in (182) are dense in Fν,T,τ (H). Let f ∈ Fν,T,τ (H) and let (ψn)n=1 be a sequence 2 of simple processes of the form in (182) that converge to f in the space Fν,T,τ (H). In a similar way to (180) we ﬁnd that Z T Z 2 E |f(s − τ, ξ)−ψn(s − τ, ξ)|H dν(ξ) ds τ E Z T −τ Z 2 = E |f(s, ξ) − ψn(s, ξ)|H dν(ξ) ds, 0 E 2 which implies that the process f(s − τ, ξ)χ(τ,T ](s) belongs to Fν,T (H). Similar 2 2 reasoning shows that f 7→ f(s−τ, ξ)χ(τ,T ](s) is bounded from Fν,T,τ (H) → Fν,T (H) with norm at most 1. As in Lemma 10.3, it is enough to prove (181) for simple functions of the form in (182) because the general case can be established using a limiting argument. If 1 2 ψ is a simple function of the form in (182), then ψ ∈ Fν,T,τ (H) ∩ Fν,T,τ (H). As a result, for t ∈ [0,T ] we have Z Z χ{τ

X Z t Z = ψ(r − τ, Ξ(r))χE0 (Ξ(r)) − ψ(r − τ, ξ) dν(ξ) dr τ E0 r∈(τ,t]∩DΞ Z Z = ψ(s − τ, ξ)χ(τ,T ](s) dπb(s, ξ). (0,t] E0 In the second step we change variables r := s + τ and in the third step we use the 1 2 fact that ψ(s − τ, ξ)χ(τ,T ](s) belongs to Fν,T (H) ∩ Fν,T (H) because it is bounded and its support has ﬁnite measure with respect to dP ⊗ ds ⊗ dν. In preparation for the piecing-out argument we give one more measurability result which gives suﬃcient conditions for the process Y (t−τ)χ{τ

Proposition 18. Let (Ω, F, P) be a probability space and let (Ft)t≥0 ⊂ F be a complete filtration. Let τ be a finite Ft-stopping time and let (Y (s))s≥0 be an E- valued stochastic process, where E is a metric space, that is adapted to the filtration (Ft+τ )t≥0. For t ≥ 0 define the stochastic process

X(t) := Y (t − τ)χ{τ

Each of the following conditions implies that X is adapted to (Ft)t≥0: i) τ takes ﬁnitely many values a.s., ii) Y is left continuous a.s., iii) Y is càdlàg a.s. and (Ft)t≥0 is right continuous.

Proof.i ) Suppose that there exist 0 ≤ s1 < s2 < ··· < sm < ∞ such that Pm τ = k=1 skχ{τ=sk} a.s. For each t ≥ 0 we may write m X X(t) = Y (t − sk)χ{τ=sk}. k=1 sk

where we deﬁne s0 := −1 for notational convenience. By assumption, Y (t−sk)

measurable with respect to the σ-field Ft−sk+τ . This means that for every r ≥ 0 we have {Y (t − sk) ∈ A} ∩ {t − sk + τ ≤ r} ∈ Fr. Take r := t in (183) to see that each set in the union on the right-hand side belongs to Ft. This shows that X(t) is Ft-measurable. ii) Now we allow τ to take infinitely many values and we assume that Y is left ∞ continuous a.s. Let (τn)n=1 be finite Ft-stopping times, each taking finitely many values, such that τn ↓ τ a.s. For each positive integer n define a stochastic process Xn by

Xn(t) := Y (t − τn)χ{τn

Since τ ≤ τn a.s. we have Ft+τ ⊆ Ft+τn for all t ≥ 0 and therefore Y is

adapted to the ﬁltration (Ft+τn )t≥0. Part i) implies that Xn is adapted to (Ft)t≥0. Since Y is left continuous a.s. we have

lim Xn(t) = Y ((t − τ)−)χ{τ

and it follows that X(t) is Ft-measurable. REVIEW OF EXISTENCE RESULTS FOR SPDES WITH LÉVY NOISE 5707 iii) Suppose now that Y is càdlàg and that the ﬁltration (Ft)t≥0 is right continuous. The reasoning from part ii) shows that the left continuous process (X(t−))t≥0 is Ft-adapted. Since the ﬁltration is right continuous it follows that X(t) = X((t−)+) is Ft-measurable.

10.2. The piecing-out argument. Now we return to the original equation (1) and allow the term K to be nonzero. We construct a solution to equation (1) using an inductive argument known as “piecing-out” in [20] and “interlacing” in [2]. We will not go through the entire argument since a similar argument has been given in [9]. Instead, we will outline the piecing-out argument and identify the places where Lemmas 10.3 and 10.4 are applied. We begin with a local solution, say (u1, τ0), to equation (83). We will show how to incorporate the first jump in the K term. Let σ1 denote the minimum of DΞ. On the event {τ0 ≤ σ1} there is nothing to do because the K term does not actually appear in (1) during the time that u1 is a solution. On the event {σ1 < τ0} we use u1 as the solution to (1) up to time σ1 and incorporate the K term at time σ1 by restarting the noise as follows. Define W (t) := W (t + σ ) − W (σ ) and Ξ(t) := Ξ(t + σ ) with domain D := f 1 1 e 1 Ξe P {s > 0 : s + σ1 ∈ DΞ}, and let π := s∈D δ . As observed above, Wf e Ξe (s,Ξ(e s)) is an Ft+σ1 -Poisson process and Ξe is an Ft+σ1 -stationary Poisson point process and the shifted filtration (Ft+σ1 )t≥0 satisfies the usual conditions. This shows that (Ω, F, (F ) , P, W, π) is a stochastic basis in the sense of Definition σ1+t t≥0 f e 4.1. Therefore, we can apply Theorem 9.2 to find the unique local pathwise solution, say (Y1, τ1), to equation (83) with respect to the shifted stochastic basis (Ω, F, (F ) , P, W, π) and with initial condition u (σ −)+K(u (σ −), Ξ(σ )). σ1+t t≥0 f e 1 1 1 1 1 Let σ2 denote the minimum of DΞ \{σ1}. We define a stochastic process u2 on the interval [0, σ2 ∧ (σ1 + τ1)) by ( u1(t) if t ∈ [0, σ1] u2(t) := Y1(t − σ1) if t ∈ [σ1, σ2 ∧ (σ1 + τ1)).

We claim that u2 solves (1) on the event {σ1 < τ} for all t ∈ [0,T ∧σ2 ∧(σ1 +τ1)). When t ∈ [0, σ1] this is true because u2(t) = u(t) for such t. When σ1 < t < T ∧ σ2 ∧ (σ1 + τ1) we have

u2(t) = Y (t − σ1) Z t−σ1 = u(σ1) − [AY (s) + B(Y (s)) + F (Y (s))] ds 0 Z t−σ1 Z Z + G(Y (s)) dWf(s) + K(Y (s−), ξ) dπeb(s, ξ) 0 (0,t−σ1] E0 Z t = u(σ1) − [AY (s − σ1) + B(Y (s − σ1)) + F (Y (s − σ1))] ds σ1 Z t

+ G(Y (s − σ1))χ(σ1,T ](s) dW (s) (184) 0 Z Z + K(Y ((s − σ )−), ξ)χ (s) dπ(s, ξ) 1 (σ1,T ] b (0,t] E0 5708 JUSTIN CYR, PHUONG NGUYEN, SISI TANG AND ROGER TEMAM

Z t = u0 − [Au2(s) + B(u2(s)) + F (u2(s))] ds 0 Z t Z Z + G(u2(s)) dW (s) + K(u2(s−)), ξ) dπb(s, ξ) 0 (0,t] E0 Z Z + K(u2(s−)), ξ) dπ(s, ξ). (0,t] E\E0 We use the change of variables Lemmas 10.3 and 10.4 in the second step of (184). This shows that u2 solves (1) on the event {σ1 < τ0} for all t ∈ [0,T ∧σ2 ∧(σ1 +τ1)). On the event {σ2 < σ1 + τ1} we incorporate the jump that occurs at time σ2. As before, we take a local solution (Y2, τ2) to equation (83) with initial condition u2(σ2−) + K(u2(σ2−), Ξ(σ2)) and we deﬁne a new process u3 to agree with u2 before time σ2 and we set u3(t) := Y2(t − σ2) for t ∈ [σ2, σ2 + τ2]. In the same way as (184), one can show that u3 solves (1) on the event {σ2 < σ1 + τ1} for all t ∈ [0,T ∧ σ3 ∧ (σ2 + τ2)). The construction continues inductively on the event {σn+1 < σn + τn} at future steps. In this way we obtain a local pathwise solution to equation (1) and establish Theorem 4.3.

Acknowledgments. This work was partially supported by the National Science Foundation (NSF) under the grant DMS-1510249 and by the Research Fund of Indiana University. The authors would like to take this opportunity to send special thanks to the reviewers and the editors for their time and eﬀorts for reviewing our paper and providing us valuable comments.

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