Spine Decomposition and Llogl Criterion for Superprocesses With

Spine decomposition and L log L criterion for superprocesses with non-local branching mechanisms

Yan-Xia Ren∗ Renming Song† and Ting Yang‡

Abstract In this paper, we provide a pathwise spine decomposition for superprocesses with both local and non-local branching mechanisms under a martingale change of measure. This result complements earlier results established for superprocesses with purely local branching mechanisms and for multitype superprocesses. As an application of this decomposition, we obtain necessary/suﬃcient conditions for the limit of the fundamental martingale to be non-degenerate. In particular, we obtain extinction properties of superprocesses with non-local branching mechanisms as well as a Kesten-Stigum L log L theorem for the fundamental martingale.

AMS 2010 Mathematics Subject Classiﬁcation: Primary 60J68, 60F15, Secondary 60F25. Keywords and Phrases: superprocess; local branching mechanism; non-local branching mechanism; spine decomposition; martingale; weak local extinction.

1 Introduction

The so-called spine decomposition for superprocesses was introduced in terms of a semigroup decomposition by Evans [22]. To be more speciﬁc, Evans [22] described the semigroup of a superprocess with branching mechanism ψ(λ) = λ2 under a martingale change of measure in terms of the semigroup of an immortal particle (called the spine) and the semigroup of

∗LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University, Beijing, 100871, P.R. China. Email: [email protected] †Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A. Email: [email protected] ‡School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R.China. Email: [email protected]

1 the original superprocess. Since then there has been a lot of interest in finding the spine decomposition for other types of superprocesses due to a variety of applications. For example, Engländerand Kyprianou [19] used a similar semigroup decomposition to establish the L1- convergence of martingales for superdiffusions with quadratic branching mechanisms. Later, Kyprianou et al. [33, 34] obtained a pathwise spine decomposition for a one-dimensional super-Brownian motion with spatially-independent local branching mechanism, in which independent copies of the original superprocess immigrate along the path of the immortal particle, and they used this decomposition to establish the Lp-boundedness (p ∈ (1, 2]) of martingales. A similar pathwise decomposition was obtained by Liu et al. [37] for a class of superdiffusions in bounded domains with spatially-dependent local branching mechanisms, and it was used to establish a Kesten-Stigum L log L theorem, which gives the sufficient and necessary condition for the martingale limit to be non-degenerate. In the set-up of branching Markov processes, such as branching diffusions and branching random walks, an analogous decomposition has been introduced and used as a tool to analyze branching Markov processes. See, for example, [25] for a brief history of the spine approach for branching Markov processes. Until very recently such a spine decomposition for superprocesses was only available for superprocesses with local branching mechanisms. In the recent paper [35], Kyprianou and Palau established a spine decomposition for a multitype continuous-state branching process (MCSBP) and used it to study the extinction properties. Concurrently to their work, a similar decomposition has been obtained by Chen et al. [12] for a multitype superdiffusion. However, in both papers, only a very special kind of non-local branching mechanisms are considered. The first goal of this paper is to close the gap by establishing a pathwise spine decomposition for superprocesses with both local and general non-local branching mechanisms. In this paper, the Schrödingeroperator associated with the mean semigroup of the superprocess is characterised by its bilinear form. Then some technical assumptions (Assumptions

1-2 below) are imposed to ensure the existence of a principal eigenvalue λ1 and a positive ground state h, and hence to ensure the existence of a fundamental martingale (Theorem 3.2 below). These assumptions may look strong, but they hold for a large class of processes, and we illustrate this for several interesting examples, including MCSBP, in Section 7. Our result (Theorem 4.6 below) shows that, for a superprocess with both local and non-local branching, under a martingale change of measure, the spine runs as a copy of a conservative process, which can be constructed by concatenating copies of a subprocess of the h-transform of the original spatial motion via a transfer kernel determined by the non-local branching mechanism, and the general nature of the branching mechanism induces three diﬀerent kinds of immigration: the continuous, discontinuous and revival-caused immigration. The concatenating procedure and revival-caused immigration are consequences of non-local branching,

2 and they do not occur when the branching mechanism is purely local. In connection with the limit theory, it is natural to ask whether or not the limit of the fundament martingale is non-degenerate. Using the spine decomposition, we establish suﬃ- cient and necessary conditions for the martingale limit to be non-degenerate, respectively, in Theorem 5.1 and Theorem 6.2. A similar idea was used in [19, 37] for (purely local branching) superdiﬀusions, and in [35] for MCSBP. However, in this paper, we extend this idea much further by considering superprocesses where the spatial motion may be discontinuous and

the branching mechanism is allowed to be generally non-local. Suppose that {Zn : n ≥ 1} is a Galton-Watson branching process with offspring distribution {pn : n ≥ 0}. Let L stand P+∞ for a random variable with this offspring distribution. Let m := n=0 npn be the mean of n the offspring distribution. Then Zn/m is a non-negative martingale. Kesten and Stigum n [30] proved that when 1 < m < +∞, the limit of Zn/m is non-degenerate if and only if E L log+ L < +∞. This result is usually referred to the Kesten-Stigum L log L theorem.

Our Corollary 6.5 shows that, in the case of λ1 < 0, the martingale limit is non-degenerate if and only if an L log L-type condition holds. This result extends an earlier result obtained in [37] for superdiffusions and can be viewed as a natural analogue of the Kesten-Stigum L log L theorem for superprocesses. Our Corollary 6.4 says that, under suitable assumptions, the non-local branching superprocess exhibits weak local extinction if and only if λ1 ≥ 0. This result can be regarded as a general non-local branching counterpart of [19, Theorem 3], where the same result is proved for a special class of superdiffusions in a domain D ⊂ Rd (the branching mechanism considered in [19] is ψ(x, λ) = −β(x)λ + α(x)λ2 with α, β being Höldercontinuous functions in D with order η ∈ (0, 1], α > 0 in D and β being bounded from above. In this paper we assume the spatial motion to be a symmetric Hunt process on a locally compact separable metric space. This assumption is not really necessary. An extension is possible. One direction is to assume the spatial motion to be a transient Borel right process on a Luzin space, whose Dirichlet form satisfies Silverstein’s sector condition. Definitions of smooth measures and Kato class can then be extended, while still preserving the properties used in this paper. We refer to [5, 14] for Kato class measures defined in this way. Neverthe- less, we keep to the less general type of spatial motions to avoid unnecessary technicalities. The rest of this paper is organized as follows. In Section 2 we review some basic definitions and properties of non-local branching superprocesses, including the definition of Kuznetsov measures which will be used later. In Section 3, we present our main working assumptions and the fundamental martingale. Section 4 provides the spine decomposition and its proof. The proof of Proposition 4.3 is postponed to the Appendix. In Sections 5 and 6 we use the spine decomposition to find sufficient and necessary conditions for the limit of the fundamental martingale to be non-degenerate respectively. In particular, we obtain

3 extinction properties of the non-local branching superprocess as well as a Kesten-Stigum L log L theorem for the martingale. In the last section, we give some concrete examples to illustrate our results. Notation and basic setting: Throughout this paper we use “:=” as a deﬁnition. We always assume that E is a locally compact separable metric space with Borel σ-algebra

B(E) and m is a σ-ﬁnite measure on (E, B(E)) with full support. Let E∂ := E ∪ {∂} be the one-point compactiﬁcation of E. Any function f on E will be automatically extended

to E∂ by setting f(∂) = 0. For a function f on E, kfk∞ := supx∈E |f(x)| and essupx∈Ef :=

infN:m(N)=0 supx∈E\N |f(x)|. Numerical functions f and g on E are said to be m-equivalent (f = g [m] in notation) if m ({x ∈ E : f(x) 6= g(x)}) = 0. If f(x, t) is a function on E ×

[0, +∞), we say f is locally bounded if supt∈[0,T ] supx∈E |f(x, t)| < +∞ for every T ∈ (0, +∞). We denote by f t(·) the function x 7→ f(x, t). Let M(E) denote the space of ﬁnite Borel measures on E topologized by the weak convergence. Let M(E)0 := M(E) \{0} where 0 denotes the null measure on E. When µ is a measure on B(E) and f, g are measurable R R functions, let hf, µi := E f(x)µ(dx) and (f, g) := E f(x)g(x)m(dx) whenever the right hand sides make sense. Sometimes we also write µ(f) for hf, µi. We use Bb(E) (respectively, B+(E)) to denote the space of bounded (respectively, non-negative) measurable functions + on (E, B(E)). For a, b ∈ R, a ∧ b := min{a, b}, a ∨ b := max{a, b}, and log a := log(a ∨ 1).

2 Preliminaries

2.1 Superprocess with non-local branching mechanisms

Let ξ = (Ω, H, Ht, θt, ξt, Πx, ζ) be an m-symmetric Hunt process on E. Here {Ht : t ≥ 0}

is the minimal admissible ﬁltration, {θt : t ≥ 0} the time-shift operator of ξ satisfying

ξt ◦ θs = ξt+s for s, t ≥ 0, and ζ := inf{t > 0 : ξt = ∂} the lifetime of ξ. Let {St : t ≥ 0} be the transition semigroup of ξ, i.e., for any non-negative measurable function f,

Stf(x) := Πx [f(ξt)] .

+ R +∞ −αt For α > 0 and f ∈ B (E), let Gαf(x) := 0 e Stf(x)dt. It is known by [6, Lemma 1.1.14] that {St : t ≥ 0} can be uniquely extended to a strongly continuous contraction 2 semigroup on L (E, m), which we also denote by {St : t ≥ 0}. By the theory of Dirichlet forms, there exists a regular symmetric Dirichlet form (E, F) on L2(E, m) associated with ξ: Z n 2 1 o F = u ∈ L (E, m) : sup (u(x) − Stu(x)) u(x)m(dx) < +∞ , t>0 t E 1 Z E(u, v) = lim (u(x) − Stu(x)) v(x)m(dx), ∀u, v ∈ F. t→0 t E

4 2 Moreover, for all f ∈ Bb(E) ∩ L (E, m) and α > 0,

Gαf ∈ F satisﬁes that Eα(Gαf, v) = (f, v) ∀v ∈ F, (2.1) where Eα(u, v) := E(u, v) + α(u, v). We assume that ξ admits a transition density p(t, x, y) with respect to the measure m, which is symmetric in (x, y) for each t > 0. Under this absolute continuity assumption, “quasi everywhere” statements can be strengthened to “everywhere” ones. Moreover, we can deﬁne notions without exceptional sets, for example, positive continuous additive functionals (PCAF in abbreviation) in the strict sense (cf. [24, Section 5.1]). In this paper, we will only deal with notions in the strict sense and omit “in the strict sense”. It is well known (see [24, Theorem A.3.21], for instance) there exist a kernel N(x, dy) on R (E, B(E)) with N(x, {x}) = 0 for all x ∈ E and a PCAF H of ξ with E Πx(Ht)µ(dx) < ∞ for all t ≥ 0 and probability measure µ on (E, B(E)) such that for any x ∈ E, any t ≥ 0, and any non-negative Borel function f on E × E vanishing on the diagonal {(y, y): y ∈ E}, " # X Z t Z Πx f(ξs−, ξs) = Πx f(ξs, y)N(ξs, dy)dHs . (2.2) s≤t 0 E

The pair (N,H) is called a L´evysystem of ξ.

In this paper, we consider a superprocess X := {Xt : t ≥ 0} with spatial motion ξ and a non-local branching mechanism ψ given by

L NL + ψ(x, f) = φ (x, f(x)) + φ (x, f) for x ∈ E, f ∈ Bb (E). (2.3)

The ﬁrst term φL in (2.3) is called the local branching mechanism and takes the form Z φL(x, λ) = a(x)λ + b(x)λ2 + e−λθ − 1 + λθ ΠL(x, dθ), x ∈ E, λ ≥ 0, (2.4) (0,+∞)

+ 2 L where a ∈ Bb(E), b ∈ Bb (E) and (θ ∧ θ )Π (x, dθ) is a bounded kernel from E to (0, +∞). The second term φNL in (2.3) is called the non-local branching mechanism and takes the form Z φNL(x, f) = −c(x)π(x, f) − 1 − e−θπ(x,f) ΠNL(x, dθ), x ∈ E, (2.5) (0,+∞) where c(x) is a non-negative bounded measurable function on E, π(x, dy) is a probability kernel on E with π(x, {x}) 6≡ 1, π(x, f) stands for R f(y)π(x, dy) and θΠNL(x, dθ) is a bounded kernel from E to (0, +∞). To be speciﬁc, X is an M(E)-valued Markov process + such that for every f ∈ Bb (E) and every µ ∈ M(E),

−hf,Xti −huf (·,t),µi Pµ e = e for t ≥ 0, (2.6)

5 −hf,Xti where uf (x, t) := − log Pδx e is the unique non-negative locally bounded solution to the integral equation Z t t−s uf (x, t) = Stf(x) − Πx ψ(ξs, uf )ds 0 Z t Z t L NL t−s = Stf(x) − Πx φ (ξs, uf (t − s, ξs))ds − Πx φ (ξs, uf )ds . (2.7) 0 0

L NL We refer to the process described above as a (St, φ , φ )-superprocess. Such a process is defined in [36] via its log-Laplace functional. Another usual way of constructing the L NL (St, φ , φ )-superprocess is as the high intensity limit of a sequence of branching particle systems, where whenever a particle dies, it chooses from two different branching types: the local branching type (when the particle dies at x, it is replaced by a random number of offspring situated at x), and the non-local branching type (when the particle dies, it gives birth to a random number of particles in E, and the offspring then start to move from their locations of birth). We refer to [36, 17] for such a construction. We define for x ∈ E, Z γ(x, dy) := c(x) + θΠNL(x, dθ) π(x, dy), γ(x) := γ(x, 1). (2.8) (0,+∞) Clearly, γ(x) is a non-negative bounded function on E and γ(x, dy) is a bounded kernel on E. Define A := {x ∈ E : γ(x) > 0}. Note that φNL(x, ·) = 0 for all x ∈ E \ A. If A = ∅ (i.e., φNL ≡ 0), we call ψ a (purely) local branching mechanism. Without loss of generality, we always assume that A 6= ∅. The arguments and results of this paper also work for (purely) local branching mechanisms. L NL It follows from [36, Theorem 5.12] that the (St, φ , φ )-superprocess has a right real- + ization in M(E). Let W0 denote the space of right continuous paths from [0, +∞) to M(E) + having zero as a trap. We may and do assume that X is the coordinate process in W0 and + that (F∞, (Ft)t≥0) is the natural filtration on W0 generated by the coordinate process. The following proposition follows from [36, Proposition 2.27 and Proposition 2.29].

Proposition 2.1. For all µ ∈ M(E) and f ∈ Bb(E),

Pµ (hf, Xti) = hPtf, µi, where Ptf(x) is the unique locally bounded solution to the following integral equation: Z t Z t Ptf(x) = Stf(x) − Πx a(ξs)Pt−sf(ξs)ds + Πx γ(ξs, Pt−sf)ds . (2.9) 0 0 + Moreover, for all µ ∈ M(E), g ∈ Bb (E) and f ∈ Bb(E),

−hg,Xti −hVtg,µi f Pµ hf, Xtie = e hVt g, µi,

6 where Vtg(x) := ug(x, t) is the unique non-negative locally bounded solution to (2.7) with f initial value g, and Vt g(x) is the unique locally bounded solution to the following integral equation Z t f f Vt g(x) = Stf(x) − Πx Ψ(ξs,Vt−sg, Vt−sg)ds , (2.10) 0 where Z Ψ(x, f, g) := g(x) a(x) + 2b(x)f(x) + θ 1 − e−f(x)θ ΠL(x, dθ) (0,+∞) Z − π(x, g) c(x) + θe−θπ(x,f)ΠNL(x, dθ) . (0,+∞)

2.2 Kuznetsov measures

L NL Let {Qt(µ, ·) := Pµ (Xt ∈ ·): t ≥ 0, µ ∈ M(E)} be the transition kernel of the (St, φ , φ )- superprocess X. Then by (2.6), we have Z −hf,νi e Qt(µ, dν) = exp (−hVtf, µi) for µ ∈ M(E) and t ≥ 0. M(E)

It implies that Qt(µ1 + µ2, ·) = Qt(µ1, ·) ∗ Qt(µ2, ·) for all µ1, µ2 ∈ M(E), and hence Qt(µ, ·)

is an inﬁnitely divisible probability measure on M(E). By the semigroup property of Qt, Vt satisﬁes that

VsVt = Vt+s for all s, t ≥ 0.

Moreover, by the inﬁnite divisibility of Qt, each operator Vt has the representation Z −hf,νi + Vtf(x) = λt(x, f) + 1 − e Lt(x, dν), t > 0, f ∈ Bb (E), (2.11) M(E)0

where λt(x, dy) is a bounded kernel on E and (1 ∧ ν(1))Lt(x, dν) is a bounded kernel from 0 0 0 E to M(E) . Let Qt be the restriction of Qt to M(E) . Let

E0 := {x ∈ E : λt(x, E) = 0 for all t > 0}.

If x ∈ E0, then we get from (2.11) that Z −hf,νi + Vtf(x) = 1 − e Lt(x, dν) for t > 0, f ∈ Bb (E). M(E)0

It then follows from [36, Proposition 2.8 and Theorem A.40] that for every x ∈ E0, the 0 family of measures {Lt(x, ·): t > 0} on M(E) constitutes an entrance law for the restricted 0 semigroup {Qt : t ≥ 0}, and hence there corresponds a unique σ-ﬁnite measure Nx on + (W0 , F∞) such that Nx({0}) = 0, and that for any 0 < t1 < t2 < ··· < tn < +∞,

0 0 Nx (Xt1 ∈ dν1,Xt2 ∈ dν2, ··· ,Xtn ∈ dνn) = Lt1 (x, dν1)Qt2−t1 (ν1, dν2) ··· Qtn−tn−1 (νn−1, dνn).

7 + It immediately follows that for all t > 0 and f ∈ Bb (E), Z −hf,Xti −hf,νi Nx 1 − e = 1 − e Lt(x, dν) = Vtf(x). (2.12) M(E)0

This measure Nx is called the Kuznetsov measure corresponding to the entrance law {Lt(x, ·): L NL t > 0} or the excursion law for the (St, φ , φ )-superprocess. When µ ∈ M(E) is sup- + ported by E0 and N(dw) is a Poisson random measure on W0 with intensity measure Z µ(dx)Nx(·), E0 the process deﬁned by Z Xe0 = µ; Xet := wtN(dw), t > 0, + W0

is a realization of the superprocess (X, Pµ). We refer to [36, section 8.4] for more details on the Kuznetsov measures. In the sequel, we assume that

Assumption 0. E+ := {x ∈ E : b(x) > 0} ⊂ E0.

Under this assumption, the Kuznetsov measure Nx exists for every x ∈ E+ when E+ is nonempty. It is established in [13] that Assumption 0 is automatically true for superdiﬀusions with a (purely) local branching mechanism. In the general case, [36, Theorem 8.6] gives the following suﬃcient condition for Assumption 0: If there is a spatially independent local branching mechanism φ(λ) taking the form Z φ(λ) = αλ + βλ2 + e−λθ − 1 + λθ n(dθ) for λ ≥ 0, (0,+∞)

where α ∈ R, β ∈ R+ and (θ ∧ θ2)n(dθ) is a bounded kernel on (0, +∞), such that φ0(λ) → +∞ as λ → +∞, and that the branching mechanism ψ of X is bounded below by φ in the sense that + ψ(x, f) ≥ φ(f(x)) for all x ∈ E and f ∈ Bb (E),

then E0 = E.

3 Fundamental martingale and weak local extinction

λ1t In this section we will establish a fundamental martingale of the form e hh, Xti for the

superprocess X in terms of the principal eigenvalue λ1 and the corresponding positive eigenfunction h of the Schrödingeroperator associated with the mean semigroup. For a MCSBP (resp. a multitype superdiffusion), if one considers the E-valued spatial motion on an en- riched state space E × I, where I is the finite or countable set of types, then the mutation

8 in types is the jumps in the I-coordinate, and the associated mean semigroup is generated by a matrix (resp. a coupled elliptic system). So, the spectral theory of matrices (resp. the potential theory for elliptic systems) can be applied. See, for example, [38, Examples 3.7 and 3.8]. For a general non-local branching superprocess considered in Subsection 2.1, the associated Schr¨odingeroperator takes the form L − a + γ, where L is the generator of underlying spatial motion, and γ is an integral operator given by γ(f)(x) = γ(x, f). Since the integral operator γ can be quite general, the method mentioned above is not applicable. Instead we characterize the Schr¨odingeroperator in terms of the associated bilinear form, and impose some technical assumptions to ensure the existence of λ1 and h.

Deﬁnition 3.1. We call a non-negative measure µ on E a smooth measure of ξ if there is µ a PCAF At of ξ such that Z Z t 1 µ + f(x)µ(dx) = lim Πm f(ξs)dAs for all f ∈ B (E). E t→0 t 0 R µ Here Πm(·) := E Πx(·)m(dx). This measure µ is called the Revuz measure of At . Moreover, we say that a smooth measure µ belongs to the Kato class K(ξ), if

Z t Z lim sup p(s, x, y)µ(dy)ds = 0. t↓0 x∈E 0 E A function q is said to be in the class K(ξ) if q(x)m(dx) is in K(ξ).

Clearly all bounded measurable functions are included in K(ξ). It is known (see, e.g., [1, Proposition 2.1.(i)] and [41, Theorem 3.1]) that if ν ∈ K(ξ), then for every ε > 0 there is some constant Aε > 0 such that Z Z 2 2 u(x) ν(dx) ≤ εE(u, u) + Aε u(x) m(dx) ∀u ∈ F. (3.1) E E R Assumption 1. E γ(x, ·)m(dx) ∈ K(ξ), where γ(x, ·) is the kernel deﬁned in (2.8). Under Assumption 1, it follows from (3.1), the boundedness of γ(x) and the inequality 1 |u(x)u(y)| ≤ (u(x)2 + u(y)2) 2 that, for every ε > 0, there is a constant Kε > 0 such that Z Z Z 2 u(x)u(y)γ(x, dy)m(dx) ≤ εE(u, u) + Kε u(x) m(dx) ∀u ∈ F. E E E It follows that the bilinear form (Q, F) deﬁned by Z Z Z Q(u, v) := E(u, v) + a(x)u(x)v(x)m(dx) − u(y)v(x)γ(x, dy)m(dx), u, v ∈ F, E E E

9 is closed and that there are positive constants K and β0 such that Qβ0 (u, u) := Q(u, u) +

β0(u, u) ≥ 0 for all u ∈ F, and

1/2 1/2 |Q(u, v)| ≤ KQβ0 (u, u) Qβ0 (v, v) ∀u, v ∈ F.

It then follows from [32] that for such a closed form (Q, F) on L2(E, m), there are unique 2 strongly continuous semigroups {Tt : t ≥ 0} and {Tbt : t ≥ 0} on L (E, m) such that β0t β0t kTtkL2(E,m) ≤ e , kTbtkL2(E,m) ≤ e , and

2 (Ttf, g) = (f, Tbtg) ∀f, g ∈ L (E, m). (3.2)

R +∞ −αt R +∞ −αt Let {Uα}α>β0 and {Ubα}α>β0 be given by Uαf := 0 e Ttfdt and Ubαf := 0 e Tbtfdt

respectively. Then {Uα}α>β0 and {Ubα}α>β0 are strongly continuous pseudo-resolvents in the sense that they satisfy the resolvent equations

Uα − Uβ + (α − β)UαUβ = 0, Ubα − Ubβ + (α − β)UbαUbβ = 0

for all α, β > β0, and

2 Qα(Uαf, g) = Qα(g, Ubαf) = (f, g) ∀f ∈ L (E, m), g ∈ F. (3.3)

L NL Recall from Proposition 2.1 that Pt is the mean semigroup of the (St, Φ , Φ )-superprocess, which satisﬁes the equation (2.9). Since γ(x, dy) is a bounded kernel on E, by (2.9), we have

for every f ∈ Bb(E),

Z t kPtfk∞ ≤ kfk∞ + (kak∞ + kγ(·, 1)k∞) kPt−sfk∞ds. 0

c1t By Gronwall’s lemma, kPtfk∞ ≤ e kfk∞ for some constant c1 > 0. For f ∈ Bb(E) and R +∞ −αt α > c1, deﬁne Rαf(x) := 0 e Ptf(x)dt. By taking Laplace transform on both sides of (2.9), we get

Rαf(x) = Gαf(x) − Gα (αRαf)(x) + Gα (γ(·,Rαf)) (x), (3.4)

2 where Gα is the α-resolvent of (St)t≥0. A particular case is when a(x), γ(x) ∈ L (E, m). 2 In this case, for all f ∈ Bb(E) ∩ L (E, m) and α suﬃciently large, both a(x)Rαf(x) and 2 γ(x, Rαf) are in Bb(E) ∩ L (E, m). Then it follows from (2.1) that Gαf, Gα (αRαf),

Gα (γ(·,Rαf)) ∈ F, and then by (3.4), (2.1) and (3.3),

Qα(Rαf, v) = (f, v) = Qα(Uαf, v) for all v ∈ F,

which implies that Rαf is m-equivalent to Uαf for α suﬃciently large. This indicates that

there is some strong relation between Pt and Tt. In fact we will show in Proposition 5.2 2 below that Ptf = Ttf [m] for every t > 0 and every f ∈ Bb(E) ∩ L (E, m). This means

10 that Pt can be regarded as a bounded linear operator on the space of bounded measurable 2 2 functions in L (E, m), which is dense in L (E, m). Hence Tt can be regarded as the unique 2 bounded linear operator on L (E, m) which is an extension of Pt.

Assumption 2. There exist a constant λ1 ∈ (−∞, +∞) and positive functions h, bh ∈ F

with h bounded continuous, khkL2(E,m) = 1 and (h, bh) = 1 such that

Q(h, v) = λ1(h, v), Q(v, bh) = λ1(v, bh) ∀v ∈ F. (3.5)

λ1t In the Theorem 3.2 below, we will prove that e hh, Xti is a non-negative martingale. To prove this we ﬁrst prove that h is invariant for some semigroup, see (3.13) below. Since

h ∈ F is continuous, it follows from [6, Theorem 4.2.6] that for every x ∈ E,Πx-a.s.

h h h(ξt) − h(ξ0) = Mt + Nt , t ≥ 0,

h h where M is a martingale additive functional of ξ having ﬁnite energy and Nt is a continuous additive functional of ξ having zero energy. The formula above is usually called Fukushima’s h decomposition. It follows from (3.5) and [24, Theorem 5.4.2] that Nt is of bounded variation, and Z t Z t Z t h Nt = −λ1 h(ξs)ds + a(ξs)h(ξs)ds − γ(ξs, h)ds, ∀t ≥ 0. 0 0 0 Following the idea of [7, Section 2], we deﬁne a local martingale on the random time interval

[0, ζp) by Z t 1 h Mt := dMs , t ∈ [0, ζp), (3.6) 0 h(ξs−) where ζp is the predictable part of the lifetime ζ of ξ, that is,  ζ if ζ < ∞ and ξζ− = ξζ , ζp = ∞, otherwise, see [40, Theorem 44.5]. Then the solution Ht of the stochastic diﬀerential equation

Z t Ht = 1 + Hs−dMs, t ∈ [0, ζp), (3.7) 0 is a positive local martingale on [0, ζp) and hence a supermartingale. Consequently, the formula h dΠx = Ht dΠx on Ht ∩ {t < ζ} for x ∈ E

h uniquely determines a family of subprobability measures {Πx : x ∈ E} on (Ω, H). Hence we have h + Πx [f(ξt)] = Πx [Htf(ξt); t < ζ] , t ≥ 0, f ∈ B (E).

11 Note that by (3.6), (3.7) and Dol´ean-Dade’s formula, 1 Y h(ξs) h(ξs) H = exp M − hM ci exp 1 − ∀t ∈ [0, ζ ), (3.8) t t 2 t h(ξ ) h(ξ ) p 0

c where M is the continuous martingale part of M. Applying Ito’s formula to log h(ξt), we

obtain that for every x ∈ E,Πx-a.s. on [0, ζ), 1 X h(ξs) h(ξs) − h(ξs−) log h(ξ ) − log h(ξ ) = M − hM ci + log − t 0 t 2 t h(ξ ) h(ξ ) s≤t s− s− Z t Z t γ(ξs, h) − λ1t + a(ξs)ds − ds. (3.9) 0 0 h(ξs) Put γ(x, h) q(x) := for x ∈ E. (3.10) h(x) By (3.8) and (3.9), we get

Z t Z t h(ξt) Ht = exp λ1t − a(ξs)ds + q(ξs)ds . 0 0 h(ξ0)

h h To emphasize, the process ξ under {Πx, x ∈ E} will be denoted as ξ . For a measurable function g, we set Z t eg(t) := exp − g(ξs)ds ∀t ≥ 0, 0 whenever it is well deﬁned. Then we have for all f ∈ B+(E) and t ≥ 0,

eλ1t Shf(x) := Πh f(ξh) = Π [e (t)h(ξ )f(ξ )] . (3.11) t x t h(x) x a−q t t

It follows from [7, Theorem 2.6] that the transformed process ξh is a conservative and recur- 2 rent (in the sense of [24]) me -symmetric right Markov process on E with me (dy) := h(y) m(dy). Thus h St 1 = 1 [me ] for all t > 0. (3.12) h h h Note that for all t > 0 and x ∈ E, the measure St (x, ·) := Πx ξt ∈ · is absolutely contin- h uous with respect to me , since St (x, ·) is absolutely continuous with respect to the measure St(x, ·) := Πx(ξt ∈ ·) by (3.11) and the latter is absolutely continuous with respect to m. Moreover, by the right continuity of the sample paths of ξh, one can easily verify that both h h 1 and St 1(x) are excessive functions for {St : t > 0}. Thus by [6, Theorem A.2.17], (3.12) implies that eλ1t 1 = Sh1(x) = Π [e (t)h(ξ )] for all x ∈ E. (3.13) t h(x) x a−q t

12 h Theorem 3.2. Suppose Assumptions 1-2 hold. Then for every µ ∈ M(E), Wt (X) := λ1t e hh, Xti is a non-negative Pµ-martingale with respect to the ﬁltration {Ft : t ≥ 0}.

Proof. By the Markov property of X, it suﬃces to prove that for all x ∈ E and t ≥ 0,

−λ1t Pth(x) = Pδx (hh, Xti) = e h(x). (3.14)

R t R t A(0,t) Let A(s, t) := − s a(ξr)dr + s q(ξr)dr and u(t, x) := Πx e h(ξt) . Clearly by (3.13), u(t, x) = e−λ1th(x). Note that

Z t A(0,t) A(t,t) A(0,t) A(s,t) e − 1 = −(e − e ) = (−a(ξs) + q(ξs)) e ds. (3.15) 0 By (3.15), Fubini’s theorem and the Markov property of ξ, we have

u(t, x) A(0,t) = Sth(x) + Πx (e − 1)h(ξt) Z t Z t A(s,t) A(s,t) = Sth(x) − Πx a(ξs)e h(ξt)ds + Πx q(ξs)e h(ξt)ds 0 0 Z t Z t γ(ξs, h) = Sth(x) − Πx a(ξs)u(t − s, ξs)ds + Πx u(t − s, ξs)ds 0 0 h(ξs) Z t Z t t−s = Sth(x) − Πx a(ξs)u(t − s, ξs)ds + Πx γ(ξs, u )ds . 0 0

In the last equality above we use the fact that u(t − s, x) = e−λ1(t−s)h(x). Thus u(t, x) is a locally bounded solution to (2.9) with initial value h. By the uniqueness of the solution, we

get u(t, x) = Pth(x) = Pδx (hh, Xti). For µ ∈ M(E), we say that the process X exhibits weak local extinction (resp. local extinction) under Pµ if for every nonempty relatively compact open subset B of E,

Pµ (limt→+∞ Xt(B) = 0) = 1 (resp. Pµ (Xt(B) = 0 for sufficitently large t) = 1). It is proved in [20] (see also [19]) that local extinction and weak local extinction coincide for superdiffusions in a domain D ⊂ Rd with local branching mechanism ψ(x, λ) = −β(x)λ + α(x)λ2, where α and β are Höldercontinuous functions on D with order η ∈ (0, 1], α > 0 in D and β is bounded from above. However the two notions are different in general. In this paper we are only concerned with weak local extinction. Corollary 3.3. Suppose Assumptions 1-2 hold. For all µ ∈ M(E) and nonempty relatively compact open subset B of E, λ1t Pµ lim sup e Xt(B) < +∞ = 1. t→+∞

In particular, if λ1 > 0, then X exhibits weak local extinction under Pµ.

13 Proof. This corollary follows immediately from Theorem 3.2 and the fact that h 1 λ1t λ1t h e Xt(B) ≤ e h 1B,Xti ≤ Wt (X). infx∈B h(x) infx∈B h(x)

Remark 3.4. Corollary 3.3 implies that the local mass of Xt grows subexponentially and the growth rate can not exceed −λ1. However, when one considers the total mass process h1,Xti, the growth rate may actually exceed −λ1. We refer to [19] and [21] for more concrete examples.

4 Spine decomposition

4.1 Concatenation process

We assume Assumptions 1-2 hold. It is well-known (see, e.g., [40, p. 286]) that for every h x ∈ E, there is a unique (up to equivalence in law) right process ((ξbt)t≥0; Πb x) on E with lifetime ζb and cemetery point ∂, such that

h h h Πb x ξbt ∈ B = Πx eq(t); ξt ∈ B ∀B ∈ B(E),

h where q is the nonnegative function deﬁned in (3.10). ξb is called the eq(t)-subprocess of ξ , h which can be obtained by killing ξ with rate q. In fact, a version of the eq(t)-subprocess can be obtained by the following method of curtailment of the lifetime. Let Z be an exponential random variable, of parameter 1, independent of ξh. Put

Z t h ζb(ω) := inf{t ≥ 0 : q ξs (ω) ds ≥ Z(ω)} (= +∞, if such t does not exist), 0 and  h ξt (ω) if t < ζb(ω), ξbt(ω) := ∂ if t ≥ ζb(ω).

h h Then the process ((ξbt)t≥0, Πx) is equal in law to the eq(t)-subprocess of ξ . Now we deﬁne a probability on E by h(y)γ(x, dy) h(y)π(x, dy) πh(x, dy) := = for x ∈ E. (4.1) γ(x, h) π(x, h)

Let ξe:= (Ωe, Ge, Get, θet, ξet, Πe x, ζe) be the right process constructed from ξb and the instantaneous distribution κ(ω, dy) := πh(ξ (ω), dy) by using the so-called “piecing out” procedure bζb(ω)− (cf. Ikeda et al. [28]). We will follow the terminology of [40, Section II.14] and call ξe a

14 concatenation process defined from an infinite sequence of copies of ξb and the transfer kernel κ(ω, dy). One can also refer to [36, Section A.6] for a summary of concatenation processes. The intuitive idea of this concatenation is described as follows. The process ξe evolves as the process ξh until time ζb, it is then revived by means of the kernel κ(ω, dy) and evolves again as ξh and so on, until a countably infinite number of revivals have occurred. Clearly in the case of purely local branching mechanism (i.e. γ(x) ≡ 0 on E), we have ζb = +∞ almost surely and hence ξe runs as a copy of ξh. Let Se t be the transition semigroup of ξe. It satisfies the following renewal equation. Z t h h h h h h + Se tf(x) = Πx eq(t)f(ξt ) + Πx q(ξs )eq(s)π (ξs , Se t−sf)ds , f ∈ Bb (E). (4.2) 0 By [36, Proposition 2.9], the above equation can be rewritten as

Z t Z t h h h h h h h h h Se tf(x) = Πx f(ξt ) − Πx q(ξs )Se t−sf(ξs )ds + Πx q(ξs )π (ξs , Se t−sf)ds . 0 0

+ Proposition 4.1. For all f ∈ Bb (E), t ≥ 0 and x ∈ E,

eλ1t Se tf(x) = Pt(fh)(x). (4.3) h(x)

In particular, Se t1(x) ≡ 1, and hence ξe has inﬁnite lifetime. Moreover, for each t > 0 and x ∈ E, ξe has a transition density pe(t, x, y) with respect to the probability measure ρ(dy) := h(y)bh(y)m(dy).

Proof. By (4.2), (3.11), (2.8), (3.10) and (4.1), we have

eλ1t Se tf(x) = Πx [ea(t)h(ξt)f(ξt)] h(x) eλ1t Z t −λ1(t−s) h + Πx ea(s)q(ξs)h(ξs)e π (ξs, Se t−sf)ds h(x) 0 eλ1t = Π [e (t)h(ξ )f(ξ )] h(x) x a t t eλ1t Z t −λ1(t−s) + Πx ea(s)γ(ξs, e hSe t−sf)ds . (4.4) h(x) 0

−λ1t Let u(t, x) := e h(x)Se tf(x). Clearly u(t, x) is a locally bounded function on [0, +∞)×E. Moreover, it follows from (4.4) and [36, Proposition 2.9] that

Z t t−s u(t, x) = Πx [ea(t)h(ξt)f(ξt)] + Πx ea(s)γ(ξs, u )ds 0 Z t Z t t−s = Πx [h(ξt)f(ξt)] − Πx a(ξs)u(t − s, ξs)ds + Πx γ(ξs, u )ds . 0 0

15 This implies that u(t, x) is a locally bounded solution to (2.9) with initial value fh. Hence

−λ1t we get e h(x)Se tf(x) = u(t, x) = Pt(fh)(x) by the uniqueness of the solution. It then follows from (3.14) that Se t1(x) ≡ 1 on E. To prove the second part of this proposition, it suﬃces to prove that for all t > 0 and x ∈ E, Se t1B(x) = 0 for all B ∈ B(E) with ρ(B) = 0 (or equivalently, m(B) = 0). Note R that Πx [h1B(ξt)] = B h(y)p(t, x, y)m(dy) = 0. It follows from the above argument that −λ1t e h(x)Se t1B(x) = Pt(h1B)(x) is the unique locally bounded solution to (2.9) with initial value 0. Thus Se t1B(x) ≡ 0.

Remark 4.2. The formula (4.3) can be written as

Pδx [hfh, Xti] h i + = Πe x f(ξet) for f ∈ Bb (E) and t ≥ 0, (4.5) Pδx [hh, Xti] which enables us to calculate the ﬁrst moment of the superprocess in terms of an auxiliary process. An analogous formula for a special class of non-local branching Markov processes, which is called a “many-to-one” formula, is established in [2], but with a totally diﬀerent method. By (3.14), we may rewrite (4.5) as h i λ1t + Pδx [hfh, Xti] = e h(x)Πe x f(ξet) for f ∈ Bb (E) and t ≥ 0.

Let τ1 be the ﬁrst revival time of ξe. For n ≥ 2, deﬁne τn recursively by τn := τn−1 + τ1 ◦

θeτn−1 . Since ξe has inﬁnite lifetime, Πe x (limn→+∞ τn = +∞) = 1 for all x ∈ E. Proposition 4.3. For all f(s, x, y), g(s, x, y) ∈ B+([0, +∞) × E × E), t > 0 and x ∈ E, we have " # Z t Z X h Πe x f(τi, ξeτi−, ξeτi ) = Πe x ds π (ξes, dy)q(ξes)f(s, ξes, y) (4.6) 0 E τi≤t and  !   X X Πe x  f(τi, ξeτi−, ξeτi )  g(τj, ξeτj −, ξeτj ) τi≤t τj ≤t " # Z t Z X h = Πe x fg(τi, ξeτi−, ξeτi ) + Πe x ds π (ξes, dy)q(ξes)f(s, ξes, y) 0 E τi≤t Z t−s Z h · Πe y dr π (ξer, dz)q(ξer)g(s + r, ξer, z) 0 E Z t Z h + Πe x ds π (ξes, dy)q(ξes)g(s, ξes, y) 0 E Z t−s Z h ·Πe y dr π (ξer, dz)q(ξer)f(s + r, ξer, z) . (4.7) 0 E The proof of this proposition will be given in the Appendix below.

16 4.2 Spine decomposition

In this section we work under Assumptions 0-2. Recall from Theorem 3.2 that the pro- h cess Wt (X) is a non-negative Pµ-martingale for every µ ∈ M(E). We can deﬁne a new 0 probability measure Qµ for every µ ∈ M(E) by the following formula:

1 h dQµ| := Wt (X)dPµ for all t ≥ 0. Ft hh, µi Ft

+ It then follows from Proposition 2.1 that for any f ∈ Bb (E) and t ≥ 0,

eλ1t eλ1t Q e−hf,Xti = P hh, X ie−hf,Xti = e−hVtf,µihV hf, µi, µ hh, µi µ t hh, µi t

h where Vt f(x) is the unique locally bounded solution to (2.10) with initial value h. In this

subsection we will establish the spine decomposition of X under Qµ. Deﬁnition 4.4. For all µ ∈ M(E) and x ∈ E, there is a probability space with probability

measure Pµ,x that carries the following processes.

(i) ((ξet)t≥0; Pµ,x) is equal in law to ξe, a copy of the concatenation process starting from x;

(ii) (n; Pµ,x) is a random measure such that, given ξe starting from x, n is a Poisson random n,t n,t measure which issues M(E)-valued processes X := (Xs )s≥0 at space-time points (ξet, t) with rate d × 2b(ξ )dt. Nξet et

Here for every y ∈ E+ = {z ∈ E : b(z) > 0}, Ny denotes the Kuznetsov measure + L NL on W0 corresponding to the (St, φ , φ )-superprocess, while for y ∈ E \ E+, Ny + denotes the null measure on W0 . Note that, given ξe, immigration happens only at n space-time points (ξet, t) with b(ξet) > 0. Let D denote the almost surely countable set n n n,t n of immigration times, and Dt := D ∩ [0, t]. Given ξe, the processes {X : t ∈ D } are mutually independent.

(iii) (m; Pµ,x) is a random measure such that, given ξe starting from x, m is a Poisson m,t m,t random measure which issues M(E)-valued processes X := (Xs )s≥0 at space-time points (ξet, t) with initial mass θ at rate

L θΠ (ξet, dθ) × dPθδ × dt. ξet

L NL m Here Pθδx denotes the law of the (St, φ , φ )-superprocess starting from θδx. Let D m m denote the almost surely countable set of immigration times, and Dt := D ∩ [0, t]. Given ξe, the processes {Xm,t : t ∈ Dm} are mutually independent, also independent of n and {Xn,t : t ∈ Dn}.

17 r,i (iv) {((Xs )s≥0; Pµ,x), i ≥ 1} is a family of M(E)-valued processes such that, given ξe r,i r,i starting from x (including its revival times {τi : i ≥ 1}), X := (Xs )s≥0 is equal L NL in law to ((Xs)s≥0, Pπi ) where Pπi denotes the law of the (St, φ , φ )-superprocess

starting from πi(·) := Θiπ(ξeτi−, ·) and Θi is a [0, +∞)-valued random variable with

distribution η(ξeτi−, dθ) given by c(x) 1 η(x, dθ) := 1 (x) + 1 (x) δ (dθ) + 1 (x)1 (θ)θΠNL(x, dθ). (4.8) γ(x) A E\A 0 γ(x) A (0,+∞)

Moreover, given ξe starting from x (including {τi : i ≥ 1}), {Θi : i ≥ 1} are mutually independent, {Xr,i : i ≥ 1} are mutually independent, also independent of n, m, {Xn,t : t ∈ Dn} and {Xm,t : t ∈ Dm}.

L NL (v) ((Xt)t≥0; Pµ,x) is equal in law to ((Xt)t≥0;Pµ), a copy of the (St, φ , φ )-superprocess starting from µ. Moreover, ((Xt)t≥0; Pµ,x) is independent of ξe, n, m and all the immigration processes. We denote by

c X n,s d X m,s r X r,i It := Xt−s,It := Xt−s and It := Xt−τi n m s∈Dt s∈Dt τi≤t the continuous immigration, the discontinuous immigration and the revival-caused immigration, respectively. We deﬁne Γt by

c d r Γt := Xt + It + It + It , ∀t ≥ 0.

c d r The process ξe is called the spine process, and the process It := It + It + It is called the immigration process. For any µ ∈ M(E) and any measure ν on (E, B(E)) with 0 < hh, νi < +∞, we randomize the law Pµ,x by replacing the deterministic choice of x with an E-valued random variable having distribution h(x)ν(dx)/hh, νi. We denote the resulting law by Pµ,ν. That is to say, 1 Z Pµ,ν(·) := Pµ,x(·)h(x)ν(dx). hh, νi E

Clearly Pµ,δx = Pµ,x. Since the laws of X and (ξ,e I) under Pµ,ν do not depend on ν and µ respectively, we sometimes write Pµ,· or P·,ν. For simplicity we also write Pµ for Pµ,µ. Here we take the convention that P0(Γt = 0 ∀t ≥ 0) = 1. For s ≥ 0, deﬁne

m m,s m m Λs := h1,X0 i, if s ∈ D and Λs := 0 otherwise. (4.9)

m L Then, given ξe, {Λs , s ≥ 0} is a Poisson point process with characteristic measure θΠ (ξes, dθ). m Let G be the σ-ﬁeld generated by ξe (including {τi : i ≥ 1}), {Θi : i ≥ 1}, {Dt : t ≥ 0}, n m {Dt : t ≥ 0}, and {Λs , s ≥ 0}.

18 0 + Proposition 4.5. For µ ∈ M(E) , f ∈ Bb (E) and t ≥ 0, we have Pµ-a.s. X Pµ [hf, Γti|G] = hPtf, µi + Pt−sf(ξes) n s∈Dt X m X + Λs Pt−sf(ξes) + Θiπ(ξeτi−, Pt−τi f). (4.10) m s∈Dt τi≤t + Proof. By (2.12), we have for every x ∈ E+ = {x ∈ E : b(x) > 0}, f ∈ Bb (E) and t > 0,

Nx (hf, Xti) = Pδx (hf, Xti) = Ptf(x).

r Let Dt := {τi : τi ≤ t}. Then by the deﬁnition of Γt, under Pµ, X n,s Pµ [hf, Γti|G] = Pµ (hf, Xti) + Pµ [hf, Xt−si|G] n s∈Dt X m,s X r,i + Pµ [hf, Xt−s i|G] + Pµ hf, Xt−si|G m r s∈Dt s=τi∈Dt X = P (hf, X i) + (hf, X i) µ t Nξes t−s n s∈Dt X X + PΛmδ (hf, Xt−si) + Pπ (hf, Xt−si) s ξes i m r s∈Dt s=τi∈Dt X = hPtf, µi + Pt−sf(ξes) n s∈Dt X m X + Λs Pt−sf(ξes) + Θiπ(ξes−, Pt−sf). m r s∈Dt s=τi∈Dt

L NL The following is our main result on the spine decomposition for the (St, φ , φ )-superprocess. Its proof will be given in the next subsection. Theorem 4.6. Suppose that Assumptions 0-2 hold. For every µ ∈ M(E)0, the process

((Γt)t≥0; Pµ) is Markovian and has the same law as ((Xt)t≥0;Qµ). Remark 4.7. In the case of a purely local branching mechanism, the revival-caused immigration does not occur. To be more speciﬁc, in that case the spine runs as a copy of the h-transformed process ξh while only continuous and discontinuous immigration occur along the spine. The concatenating procedure and the revival-cased immigration are consequences of non-local branching. Similar phenomenon has been observed in [35] for multitype continuous-state branching processes and in [12] for multitype superdiﬀusions. Remark 4.8. The non-local branching mechanism ψ given by (2.3)-(2.5) is not the most general form that can be assumed to establish a spine decomposition. In fact, we can establish a spine decomposition for the class of branching mechanisms developed in [17]:

L NL + ψ(x, f) = φ (x, f(x)) + φ (x, f) for x ∈ E, f ∈ Bb (E),

19 where φL takes the same form of (2.4) and Z Z Z φNL(x, f) = − c(x, π)π(f)G(x, dπ) − 1 − e−θπ(f) ΠNL(x, π, dθ)G(x, dπ), P(E) P(E) (0,+∞) where P(E) denotes the space of probability measures on E, G(x, dπ) is a probability measure from E to P(E), c(x, π) is a nonnegative bounded measurable function on E × P(E), and θΠNL(x, π, dθ) is a bounded kernel from E × P(E) to (0, +∞). It is easy to see that φNL has the form given in (2.5) when G(x, dπ) is a Dirac measure on P(E). To establish the spine decomposition, one should redeﬁne γ(x, dy) as Z γ(x, dy) = r(x, π)π(dy)G(x, dπ), P(E)

R NL where r(x, π) := c(x, π) + (0,+∞) θΠ (x, π, dθ). As a result, the instantaneous distribution, deﬁned by (4.1), of the concatenation process (the spine) ξe, changes with γ accordingly. Regarding the spine decomposition for the above branching mechanism, there is a new feature of the revival-caused immigration, which is described as follows: Given the spine ξe(including

its revival times), at each revival time τi, a probability measure πi is chosen from P(E), ∗ independently, according to the distribution G (ξeτi−, dπ), where π(h)r(x, π)G(x, dπ) G∗(x, dπ) := . γ(x, h)

r,i An immigration (Xs )s≥0 then occurs at τi, and it is equal in law to the process ((Xs)s≥0, PΘiπi ),

where Θi is an independent [0, +∞)-valued random variable with distribution η(ξeτi−, πi, dθ) given by c(x, π) η(x, π, dθ) := 1 + 1 δ (dy) r(x, π) {r(x,π)>0} {r(x,π)=0} 0 θ1 ΠNL(x, π, dθ) + {θ∈(0,+∞)} 1 . r(x, π) {r(x,π)>0} We omit the details of the proof here for brevity.

4.3 Proof of Theorem 4.6

In this subsection, we give the proof of Theorem 4.6. In order to do this, we prove a few lemmas ﬁrst.

+ Lemma 4.9. For all x ∈ E, t ≥ 0 and f ∈ Bb (E), Z t h c d i P·, x exp −hf, It + It i | ξes : 0 ≤ s ≤ t = exp − Φ(ξes,Vt−sf(ξes))ds , 0

R −λθ L where Φ(x, λ) := 2b(x)λ + (0,+∞) θ 1 − e Π (x, dθ) for x ∈ E and λ ≥ 0.

20 Proof. This lemma follows from an argument which is almost identical to the one leading to (59)–(60) in [33]. We omit the details here.

+ Lemma 4.10. Suppose f, l ∈ Bb (E) and (x, s) 7→ gs(x) is a non-negative locally bounded measurable function on E × [0, +∞). For all x ∈ E and t > 0, let

Z t −w(x,t) r e := P·, x exp − gt−s(ξes)ds − hf, It i − l(ξet) . 0

Then u(t, x) := e−λ1th(x)e−w(x,t) satisﬁes the following integral equation:

Z t −l(ξt) u(t, x) = Πx e h(ξt) + Πx ds Φ(ξs,Vt−sf(ξs))u(t − s, ξs) 0 t−s − Ψ(ξs,Vt−sf, u ) − gt−s(ξs)u(t − s, ξs) , (4.11) where Ψ and Φ are deﬁned in Proposition 2.1 and Lemma 4.9 respectively.

Proof. Following the idea of [23], it suﬃces to prove the result in the case when g does not depend on the time variable. Let τ1 denote the ﬁrst revival time of ξe. We have the following fundamental equation:

t h h i Z −w(x,t) h −l(ξt ) h h e = Πx eq+g(t)e + Πx ds q(ξs )eq+g(s) 0

Z h h h −wt−s −θπ(ξs ,Vt−sf) h · π (ξs , e ) e η(ξs , dθ) . [0,+∞)

The ﬁrst term corresponds to the case when τ1 ≥ t, and the second term corresponds to the case when the ﬁrst revival happens at time s ∈ (0, t). It then follows from Fubini’s theorem and (3.11) that

Z t −λ1t −w(x,t) −l(ξt) e h(x)e = Πx ea+g(t)h(ξt)e + Πx ds ea+g(s)q(ξs)h(ξs) 0 Z h −λ1(t−s) −wt−s −θπ(ξs,Vt−sf) · π (ξs, e e ) e η(ξs, dθ) . [0,+∞)

We continue the above calculation by [36, Proposition 2.9] and (4.8) to get

u(t, x) = e−λ1th(x)e−w(x,t) Z t −l(ξt) −λ1(t−s) −w(ξs,t−s) = Πx h(ξt)e − Πx (a(ξs) + g(ξs))e h(ξs)e ds 0 Z t Z −λ1(t−s) h −wt−s −θπ(ξs,Vt−sf) + Πx ds q(ξs)h(ξs)e π (ξs, e ) e η(ξs, dθ) 0 [0,+∞)

21 Z t −l(ξt) −λ1(t−s) −w(ξs,t−s) = Πx h(ξt)e − Πx (a(ξs) + g(ξs))e h(ξs)e ds 0 Z t Z −λ1(t−s) −wt−s −θπ(ξs,Vt−sf) NL + Πx ds π(ξs, e he ) c(ξs) + re Π (ξs, dθ) . 0 (0,+∞)

This directly leads to (4.11).

+ Lemma 4.11. For all f, g ∈ Bb (E), µ ∈ M(E), x ∈ E and t ≥ 0,

λ1t h i −g e −hVtf,µi he µ,x exp −hf, Γti − g(ξet) = e V f(x), (4.12) P h(x) t

he−g −g where Vt f(x) is the unique locally bounded solution to (2.10) with initial value he .

Proof. Recall from Deﬁnition 4.4 that (X; Pµ,x) is independent of ξe and all the immigration r c d processes. Moreover, given ξe (including {τi : i ≥ 1}), I is independent of I and I . It then follows from Lemma 4.9 that h i Pµ,x exp −hf, Γti − g(ξet)

h c d r i = Pµ,x exp −hf, Xti − hf, It + It i − hf, It i − g(ξet) n h i −hf,Xti −g(ξet) c d = Pµ,x e Pµ,x e Pµ,x exp −hf, It + It i | ξes : 0 ≤ s ≤ t

h r io ·Pµ,x exp (−hf, It i) |{ξes : 0 ≤ s ≤ t} ∪ {τi : τi ≤ t} Z t −hVtf,µi r = e P·, x exp − Φ(ξes,Vt−sf(ξes))ds − hf, It i − g(ξet) . (4.13) 0 h i −λ1t R t r Let v(t, x) := e h(x)P·, x exp − 0 Φ(ξes,Vt−sf(ξes))ds − hf, It i − g(ξet) . One can easily verify that (x, s) 7→ gs(x) := Φ(x, Vsf(x)) is a locally bounded function. Thus by Lemma 4.10, v(t, x) is a locally bounded solution to the equation (2.10) with initial value he−g. By he−g the uniqueness of such a solution, we have v(t, x) = Vt f(x). This and (4.13) lead to (4.12).

0 Proof of Theorem 4.6: First we claim that for every µ ∈ M(E) , ((Γt)t≥0; Pµ) has the + same one dimensional distribution as ((Xt)t≥0;Qµ). This would follow if for every f ∈ Bb (E) and every t ≥ 0,

−hf,Γti −hf,Xti Pµ e = Qµ e . (4.14)

By the deﬁnition of Qµ and Proposition 2.1,

eλ1t eλ1t Q e−hf,Xti = P hh, X ie−hf,Xti = e−hVtf,µihV hf, µi, (4.15) µ hh, µi µ t hh, µi t

22 h where Vt f(x) is the unique locally bounded solution to (2.10) with initial value h. By Lemma 4.11, we have

−1 h Pµ,x [exp (−hf, Γti)] = exp (λ1t − hVtf, µi) h(x) Vt f(x).

Thus 1 Z −hf,Γti −hf,Γti Pµ e = Pµ,x e h(x)µ(dx) hh, µi E eλ1t = e−hVtf,µihV hf, µi. (4.16) hh, µi t

Combining (4.15) and (4.16), we get (4.14). It follows that for every µ ∈ M(E)0,

Pµ(Γt = 0) = Qµ(Xt = 0) 1 = P W h(X); X = 0 = 0 ∀t > 0. (4.17) hh, µi µ t t

It remains to prove the Markov property of ((Γt)t≥0; Pµ). To do this, we apply [23, Lemma 3.3] here. Recall that E∂ = E ∪ {∂} where ∂ is a cemetery point. We can extend the probability measure Pµ,x onto µ × {∂} by deﬁning that Pµ,∂(ξet = ∂, It = 0 ∀t ≥ 0) = 1 for all µ ∈ M(E). In the remainder of this proof, we call J a Markov kernel if J is a map from the measurable space (S, S) to the measurable space (S0, S0) such that for every y ∈ S, J(y, ·) is a probability measure on (S0, S0), and for every B ∈ S0, J(·,B) ∈ bS the space of bounded measurable functions on S. The kernel J will also be viewed as an operator taking 0 R f ∈ bS to Jf ∈ bS where Jf(y) := S0 f(z)J(y, dz). Clearly ((Zt)t≥0 := ((Γt, ξet))t≥0; Pµ,x) is a Markov process on M(E) × E∂. Denote by St the transition semigroup of Zt, by K the Markov kernel from M(E) × E∂ to M(E) induced

by the projection from M(E) × E∂ onto M(E), and by Q the Markov kernel from M(E)

to M(E) × E∂ given by

h(x)ν1(dx) Q(ν1, d(ν2 × x)) := 1{ν16=0}δν1 (dν2) × 1E(x) + 1{ν1=0}δ0(dν2) × δ∂(dx). hh, ν1i

Let Rt := QStK for t ≥ 0. One can easily verify that QK is the identity kernel on M(E)

and Rt(ν1, dν2) = Pν1 (Γt ∈ dν2) for all ν1 ∈ M(E). By [23, Lemma 3.3], ((Γt)t≥0; Pµ) is + Markovian as long as QSt = RtQ. This would follow if for all f, g ∈ Bb (E) and ν1 ∈ M(E), Z Z −hf,ν3i−g(y) e Q(ν2, d(ν3 × y))Rt(ν1, dν2) M(E) M(E)×E∂ Z Z −hf,ν3i−g(y) = e St(ν2 × x, d(ν3 × y))Q(ν1, d(ν2 × x)). (4.18) M(E)×E∂ M(E)×E∂

23 By the above deﬁnitions, we have

−g −hf,Γti hhe , Γti LHS of (4.18) = Pν1 e 1{Γt6=0} + Pν1 (Γt = 0) , hh, Γti h i −hf,Γti−g(ξet) RHS of (4.18) = Pν1 e 1{ν16=0} + 1{ν1=0}.

0 + In view of (4.17), to show (4.18), it suﬃces to show that for all µ ∈ M(E) and f, g ∈ Bb (E), h i hhe−g, Γ i −hf,Γti−g(ξet) −hf,Γti t Pµ e = Pµ e 1{Γt6=0} . (4.19) hh, Γti It follows from Lemma 4.11 that h i 1 Z h i −hf,Γti−g(ξet) −hf,Γti−g(ξet) Pµ e = Pµ,x e h(x)µ(dx) hh, µi E λ1t e −g = e−hVtf,µihV he f, µi, (4.20) hh, µi t

he−g −g where Vt f(x) is the unique locally bounded solution to (2.10) with initial value he . On the other hand, since (Γt, Pµ) and (Xt, Qµ) are identically distributed for each t ≥ 0, we have by the deﬁnition of Qµ and Proposition 2.1 that

−g −g −hf,Γti hhe , Γti −hf,Xti hhe ,Xti Pµ e 1{Γt6=0} = Qµ e 1{Xt6=0} hh, Γti hh, Xti eλ1t = P e−hf,Xtihhe−g,X i hh, µi µ t

λ1t e −g = e−hVtf,µihV he f, µi. (4.21) hh, µi t

Combining (4.20) and (4.21), we get (4.19). The proof is now complete.

5 Suﬃcient conditions for a non-degenerate martingale limit

In this section, we will give suﬃcient conditions for the fundamental martingale to have a non-degenerate limit. We start with an assumption. Assumption 3.

(i) Either one of the following conditions holds.

(1) a(x), γ(x) ∈ L2(E, m).

24 (2) The L´evysystem (N,H) of ξ is of the form (N, t), where N is given by

N(x, dy) = N(x, y)m(dy)

with N(x, y) being a symmetric Borel function on E × E. The probability kernel π(x, dy) has a density π(x, y) with respect to the measure m such that

γ(x)π(x, y) = F (x, y)N(x, y) ∀x, y ∈ E

for some non-negative bounded Borel function F (x, y) on E × E vanishing on the diagonal.

(ii) (1Aπ(·, h), bh) < +∞.

(iii) x 7→ π(x, h)/h is bounded from above on A.

It is easy to see that Assumption 3.(iii) implies Assumption 3.(ii). In this section we will use the ﬁrst two items of this assumption. In the next section we will use items (i) and (iii) of this assumption. The following theorem, giving an L log L type criterion for the martingale limit to be non-degenerate, is the main result of this section.

h Theorem 5.1. Suppose Assumptions 0–2 and 3.(i)–(ii) hold. Let W∞(X) be the almost sure h limit of the non-negative martingale Wt (X). Suppose that Z rh(·) log+(rh(·))ΠL(·, dr), bh (0,+∞) Z + rπ(·, h) log+(rπ(·, h))ΠNL(·, dr), bh < +∞. (5.1) (0,+∞) We have

h h 1 (i) if λ1 < 0, then Wt (X) converges to W∞(X) as t → ∞ in L (Pµ) for every µ ∈ M(E), h h 0 and W∞(X) is non-degenerate in the sense that Pµ(W∞(X) > 0) > 0 for µ ∈ M(E) ;

h (ii) if λ1 > 0, then W∞(X) = 0 Pµ-a.s. for every µ ∈ M(E). In the remainder of this section we will assume Assumptions 0-2 hold. Additional conditions used are stated explicitly. To prove Theorem 5.1, we need a few preliminary results.

2 Proposition 5.2. Suppose Assumption 3.(i) holds. For all f ∈ Bb(E) ∩ L (E, m) and s, t ∈ (0, +∞), 2 lim Psf = Ptf in L (E, m). (5.2) s→t Moreover,

Ptf = Ttf [m] for all t > 0, (5.3) where (Tt)t≥0 is the semigroup associated with the bilinear form (Q, F) by (3.3).

25 Proof. First we suppose Assumption 3.(i2) holds. We note that γ(x, dy) = γ(x)π(x, dy) = γ(x)π(x, y)m(dy). Then by Assumptions 1 and 3.(i2), Z Z F (x, y)N(x, y)m(dx) = γ(x)π(x, y)m(dx) ∈ K(ξ). E E

Thus F is in the class J deﬁned in [14]. Let Fˆ(x, y) := F (y, x). Then Z Z Z Fˆ(x, y)N(x, y)m(dx) = F (y, x)N(y, x)m(dx) = γ(y)π(y, x)m(dx) = γ(y). E E E

Since γ is a bounded function on E, the above equation implies that Fˆ is in the class J. One can also show easily that the functions log(1 + F ) and log(1 + Fˆ) are in J. Deﬁne

Z t X As,t := − a(ξr)dr + log (1 + F (ξr−, ξr)) ∀0 ≤ s < t < +∞. s s

It follows from [14, Theorem 4.8] (see also [15, p. 275]) that the semigroup corresponding to the bilinear form (Q, F) deﬁned by (3.3) is

A0,t + Ttf(x) = Πx e f(ξt) ∀t ≥ 0, x ∈ E, f ∈ B (E).

p Furthermore, for any 1 ≤ p ≤ +∞,(Tt)t≥0 is semigroup on L (E, m), and for 1 ≤ p < +∞, p (Tt)t≥0 is strongly continuous semigroup on L (E, m). Similar to [14, (2.6)], we have

Z t A0,t As,t X As,t e − 1 = − e a(ξs)ds + e F (ξs−, ξs). 0 s≤t

Using this, the Markov property of ξ, (2.2) and Assumption 3.(i2), one can show that for

any f ∈ Bb(E) and x ∈ E, Z t Z t Z Ttf(x) = Πx [f(ξt)] − Πx a(ξs)Tt−sf(ξs)ds + Πx Tt−sf(y)F (ξs, y)N(ξs, y)m(dy)ds 0 0 E Z t Z t Z = Πx [f(ξt)] − Πx a(ξs)Tt−sf(ξs)ds + Πx Tt−sf(y)γ(ξs)π(ξs, y)m(dy)ds 0 0 E Z t Z t Z = Πx [f(ξt)] − Πx a(ξs)Tt−sf(ξs)ds + Πx Tt−sf(y)γ(ξs, dy)ds . 0 0 E

This implies that Ttf(x) satisﬁes the integral equation (2.9). By uniqueness Ttf(x) = Ptf(x), and thus we conclude the result of this proposition. 2 Now we suppose Assumption 3.(i1) holds. Fix f ∈ Bb(E) ∩ L (E, m). We ﬁrst prove

(5.2). Without loss of generality, we assume 0 < s < t < +∞. Let Fr(x) := −a(x)Prf(x) + c1r γ(x, Prf). We have shown in the argument below (3.3) that kPrfk∞ ≤ e kfk∞ for some c1r constant c1 > 0. Thus by deﬁnition, |Fr(x)| ≤ (|a(x)| + γ(x))kPrfk∞ ≤ e kfk∞(|a(x)| +

26 γ(x)). Clearly by the boundedness of a(x) and γ(x), (x, r) 7→ Fr(x) is locally bounded on 2 E × [0, +∞) and by Assumption 3.(i), x 7→ Fr(x) ∈ Bb(E) ∩ L (E, m). By (2.9), we have Z t Ptf(x) − Psf(x) = Stf(x) − Ssf(x) + Πx Fr(ξt−r)dr s Z s + Πx Fr(ξt−r) − Fr(ξs−r)dr . (5.4) 0 2 Recall that {St : t ≥ 0} is a strongly continuous contraction semigroup on L (E, m). Thus

kStf − SsfkL2(E,m) = kSs (St−sf − f) kL2(E,m)

≤ kSt−sf − fkL2(E,m) → 0 as s → t. (5.5)

Note that Z t Z t Z t

Πx Fr(ξt−r)dr ≤ |Πx [Fr(ξt−r)]| dr = |St−rFr(x)| dr. (5.6) s s s

We have by Minkowski’s integral inequality and the contractivity of St that Z t Z t Z t k |St−rFr| drkL2(E,m) ≤ kSt−rFrkL2(E,m) dr ≤ kFrkL2(E,m)dr s s s Z t c1r ≤ kfk∞(kakL2(E,m) + kγkL2(E,m)) e dr → 0 s as s → t. This together with (5.6) implies that Z t

lim Πx Fr(ξt−r)dr = 0. (5.7) s→t s L2(E,m) Note that by the Markov property of ξ, Z s

Πx Fr(ξt−r) − Fr(ξs−r)dr 0 Z s ≤ Πx Πξs−r (Fr(ξt−s)) − Πξs−r (Fr(ξ0)) dr 0 Z s = Ss−r (|St−sFr − Fr|)(x)dr. (5.8) 0

It follows from the strong continuity and contractivity of the semigroup {St : t ≥ 0} that

lim kSt−sFr − FrkL2(E,m) = 0, and s→t

c1r kSt−sFr − FrkL2(E,m) ≤ 2kFrkL2(E,m) ≤ 2e kfk∞(kakL2(E,m) + kγkL2(E,m)).

Thus by Minkowski’s integral inequality and the dominated convergence theorem, we have Z s Z s

Ss−r (|St−sFr − Fr|) dr ≤ kSs−r (|St−sFr − Fr|)kL2(E,m) dr 0 L2(E,m) 0

27 Z s ≤ kSt−sFr − FrkL2(E,m) dr 0 → 0 as s → t.

This together with (5.8) implies that

Z s

lim Πx Fr(ξt−r) − Fr(ξs−r)dr = 0. (5.9) s→t 0 L2(E,m)

Combining (5.4)–(5.9), we arrive at (5.2). To prove (5.3), it suﬃces to prove that for every t > 0 and every g ∈ L2(E, m), Z Z Ptf(x)g(x)m(dx) = Ttf(x)g(x)m(dx). (5.10) E E Note that by H¨older’sinequality and (5.2), for s, t ∈ (0, +∞), Z Z

Ptf(x)g(x)m(dx) − Psf(x)g(x)m(dx) E E

≤ kPtf − PsfkL2(E,m)kgkL2(E,m) → 0 (5.11) R as s → t. This implies t 7→ E Ptf(x)g(x)m(dx) is a continuous function on (0, +∞). Sim- 2 ilarly, using the strong continuity of {Tt : t ≥ 0} on L (E, m), one can prove that t 7→ R E Ttf(x)g(x)m(dx) is also a continuous function on (0, +∞). By taking the Laplace trans- R R R form of E Ptf(x)g(x)m(dx) (respectively, E Ttf(x)g(x)m(dx)), we get E Rαf(x)g(x)m(dx) R (respectively, E Uαf(x)g(x)m(dx)). It has been shown in the argument below (3.3) that under Assumption 3.(i), Rαf = Uαf [m] for α suﬃciently large. So the Laplace transforms of both sides of (5.10) are identical for α suﬃciently large. Hence (5.10) follows from Post’s inversion theorem for Laplace transforms. Proposition 5.3. Under the assumptions of Proposition 5.2, the measure

ρ(dx) = h(x)bh(x)m(dx) is an invariant probability measure for the semigroup {Se t : t ≥ 0}, i.e., for all t ≥ 0 and f ∈ B+(E), Z Z Se tf(x)ρ(dx) = f(x)ρ(dx). (5.12) E E + Proof. By the monotone convergence theorem, we only need to prove (5.12) for f ∈ Bb (E). + 2 Clearly fh ∈ Bb (E) ∩ L (E, m). It follows by (4.3),(5.3) and (3.2) that Z Z λ1t Se tf(x)ρ(dx) = e Pt(fh)(x)bh(x)m(dx) E E

28 Z λ1t = e Tt(fh)(x)bh(x)m(dx) E Z λ1t = e f(x)h(x)Tbtbh(x)m(dx) E Z = f(x)ρ(dx). E

−1 h Lemma 5.4. The function g(x) := h(x) Pδx W∞(X) satisﬁes that

h Pµ W∞(X) = hgh, µi for all µ ∈ M(E). (5.13)

Moreover,

Se tg(x) = g(x) for all t ≥ 0 and x ∈ E. (5.14)

Proof. To prove the ﬁrst claim, we note that for an arbitrary constant λ > 0, by the bounded convergence theorem,

h h Pµ exp −λW∞(X) = lim Pµ exp −λWt (X) t→+∞

= lim exp (−hlλ(t, ·), µi) t→+∞ = exp − lim hlλ(t, ·), µi , (5.15) t→+∞

h where lλ(t, x) := − log Pδx exp −λWt (X) . Let

h lλ(x) := lim lλ(t, x) = − log Pδx exp −λW∞(X) . t→+∞ We have by Jensen’s inequality that

h λ1t lλ(t, x) ≤ λPδx Wt (X) = λe Pth(x) = λh(x) for all x ∈ E, t ≥ 0.

Hence lλ(x) ≤ λh(x) for all x ∈ E. This together with (5.15) and the dominated convergence theorem yields that

h −hlλ,µi Pµ exp −λW∞(X) = e . (5.16) Thus we get (5.13) by diﬀerentiating both sides of (5.16) with respect to λ and then letting λ ↓ 0. Note that 0 ≤ g ≤ 1 by Fatou’s lemma. By the Markov property of X and (5.13), we have for all t ≥ 0 and x ∈ E, 1 λ1(t+s) g(x) = Pδ lim e hh, Xt+si h(x) x s→+∞

λ1t λ1t e h e h = Pδx PXt lim Ws (X) = Pδx PXt W∞(X) h(x) s→+∞ h(x)

29 eλ1t eλ1t = Pδx [hgh, Xti] = Pt(gh)(x) = Se tg(x). h(x) h(x) Here we used (4.3) in the last equality.

m m Lemma 5.5. Suppose Assumption 3.(i) holds. Let D and Λs be as in Deﬁnition 4.4.(iii) and (4.9) respectively. If condition (5.1) holds, then for m-almost every x ∈ E,

+ m + log (Λs h(ξes)) log Θiπ(ξeτi−, h) lim = lim = 0 ·, x-a.s. (5.17) m P D 3s→+∞ s i→+∞ τi Proof. To prove (5.17), it suﬃces to prove that for any ε > 0 suﬃciently small,

! +∞ ! X X ·, x 1 m εs = +∞ and ·, x 1 ετi = +∞ = 0. (5.18) P {Λs h(ξes)>e } P {Θiπ(ξeτi−,h)>e } s∈Dm i=1

For any B ∈ B(E) with 0 < m(B) < +∞, let µB(dx) := bh(x)1B(x)m(dx). Clearly µB ∈ 0 m M(E) . Recall that given ξe, {Λs : s ≥ 0} is a Poisson point process with characteristic L measure λΠ (ξes, dλ). Thus by Fubini’s theorem and the fact that ρ(dx) = h(x)bh(x)m(dx) is an invariant measure for Se t, we have ! X µ 1 m εs P B {Λs h(ξes)>e } s∈Dm Z +∞ Z L = λ1 εs Π (ξ , dλ)ds PµB {λh(ξes)>e } es 0 (0,+∞) Z Z +∞ Z 1 L = λ1 εs Π (ξ , dλ)ds ρ(dx) PµB ,x {λh(ξes)>e } es hh, µBi B 0 (0,+∞) Z +∞ Z Z 1 L ≤ ds ρ(dx) λ1{λh(x)>eεs}Π (x, dλ) hh, µBi 0 E (0,+∞) + ! 1 Z Z log λh(x)/ε = λΠL(·, dλ) ds, hbh hh, µBi (0,+∞) 0 1 Z = λh(·) log+(λh(·))ΠL(·, dλ), bh . (5.19) εhh, µBi (0,+∞) The right hand side of (5.19) is ﬁnite by (5.1). Thus we get ! X µ 1 m εs < +∞ = 1. P B {Λs h(ξes)>e } s∈Dm Note that ! Z ! X −1 X µ 1 m εs < +∞ = ρ(B) ·, x 1 m εs < +∞ ρ(dx). P B {Λs h(ξes)>e } P {Λs h(ξes)>e } s∈Dm B s∈Dm

30 P Thus ·, x m 1 m εs < +∞ = 1 for m-almost every x ∈ B. Since B is arbitrary, P s∈D {Λs h(ξes)>e } the ﬁrst equality of (5.18) holds for m-almost every x ∈ E.

Recall from Deﬁnition 4.4 that given ξe (including {τi : i ≥ 1}), Θi is distributed as

η(ξeτi−, dθ) given by (4.8). Thus by Fubini’s theorem and (4.6),

+∞ ! X µB 1 ετi P {Θiπ(ξeτi−,h)>e } i=1 " +∞ # X Z = PµB η(ξeτi−, dθ) ετi i=1 θπ(ξeτi−,h)>e " +∞ Z # X 1 NL = PµB 1A(ξeτi−) θΠ (ξeτi−, dθ) ετi i=1 γ(ξeτi−) θπ(ξeτi−,h)>e "Z +∞ Z # 1 NL = PµB q(ξes) 1A(ξes)ds θΠ (ξes, dθ) εs 0 γ(ξes) θπ(ξes,h)>e Z +∞ Z " Z # 1 π(ξes, h) NL = ds Pµ,x 1A(ξes) θΠ (ξes, dθ) ρ(dx) hh, µ i εs B 0 B h(ξes) θπ(ξes,h)>e Z +∞ Z Z 1 NL ≤ ds 1A(x)π(x, h)bh(x)m(dx) θΠ (x, dθ) hh, µBi 0 E θπ(x,h)>eεs log+(θπ(x,h)) Z Z Z ε 1 NL = 1A(x)π(x, h)bh(x)m(dx) θΠ (x, dθ) ds hh, µBi E (0,+∞) 0 Z 1 + NL = π(·, h) θ log (θπ(·, h))Π (·, dθ), 1Abh . (5.20) εhh, µBi (0,+∞)

The right hand side of (5.20) is ﬁnite by (5.1). Thus we get

+∞ ! X µB 1 ετi < +∞ = 1. P {Θiπ(ξeτi−,h)>e } i=1 Using an argument similar to that at the end of the ﬁrst paragraph of this proof, one can prove that the second equality of (5.18) holds for m-almost every x ∈ E. Proof of Theorem 5.1: Recall that, by assumption, Assumptions 0-2 and 3.(i)-(ii) hold. 0 h (i) Suppose λ1 < 0. Without loss of generality, we assume µ ∈ M(E) . Since Wt (X) is a non-negative martingale, to show it is a closed martingale, it suﬃces to prove

h Pµ W∞(X) = hh, µi. (5.21)

First we claim that (5.21) is true for µB(dy) := 1B(y)bh(y)m(dy) with B ∈ B(E) and 0 < m(B) < +∞. It is straightforward to see from the change of measure methodology (see, for

31 example, [18, Theorem 5.3.3]) that the proof for this claim is complete as soon as we can show that h QµB lim sup Wt (X) < +∞ = 1. (5.22) t→+∞

Since ((Xt)t≥0;QµB ) is equal in law to ((Γt)t≥0; PµB ), (5.22) is equivalent to that h PµB lim sup Wt (Γ) < +∞ = 1. (5.23) t→+∞

In the remainder of this proof, we deﬁne a function log∗ θ := θ/e if θ ≤ e and log∗ θ := log θ if θ > e. Under the assumptions of Theorem 5.1, one can prove by elementary computation that (5.1) implies

Z rh(·) log∗(rh(·))ΠL(·, dr), bh (0,+∞) Z ∗ NL + rπ(·, h) log (rπ(·, h))Π (·, dr), 1Abh < +∞. (5.24) (0,+∞)

m n Recall that G is the σ-ﬁeld generated by ξe(including {τi : i ≥ 1}), {Dt : t ≥ 0}, {Dt : t ≥ 0}, m {Θi : i ≥ 1} and {Λs : s ≥ 0}. By (4.10), for any t > 0,

h PµB Wt (Γ) | G  

λ1t X X m X = e hPth, µBi + Pt−sh(ξes) + Λs Pt−sh(ξes) + Θiπ(ξeτi−, Pt−τi h) n m s∈Dt s∈Dt τi≤t

X λ1s X λ1s m X λ1τi = hh, µBi + e h(ξes) + e Λs h(ξes) + e Θiπ(ξeτi−, , h) n m s∈Dt s∈Dt τi≤t +∞ X λ1s X λ1s m X λ1τi ≤ hh, µBi + e h(ξes) + e Λs h(ξes) + e Θiπ(ξeτi−, , h). (5.25) s∈Dn s∈Dm i=1

We begin with the second term on the right hand side of (5.25). Let ε ∈ (0, −λ1) be an arbitrary constant.

X λ1s X λ1s X λ1s e h(ξ ) = e h(ξ )1 εs + e h(ξ )1 εs =: I + II. es es {h(ξes)>e } es {h(ξes)≤e } s∈Dn s∈Dn s∈Dn P Recall that given ξe, the random measure s∈Dn δs(·) on [0, +∞) is a Poisson random measure with intensity 2b(ξet)dt, and that ρ(dx) = h(x)bh(x)m(dx) is an invariant probability measure for Se t. We have by Fubini’s theorem, " !#

X λ1s µ (II) = µ µ e h(ξs)1 εs ξr : r ≥ 0 P B P B P B e {h(ξes)≤e } e s∈Dn

32 Z +∞ λ1s = 2b(ξ )e h(ξ )1 εs ds PµB es es {h(ξes)≤e } 0 2 Z +∞ Z h i λ1s εs = e ds Pµ,x b(ξes)h(ξes)1{h(ξes) ≤ e } ρ(dx) hh, µBi 0 B 2 Z +∞ Z λ1s ≤ e ds b(x)h(x)1{h(x)≤eεs}ρ(dx) hh, µBi 0 E Z +∞ Z 2kbk∞ ≤ e(λ1+ε)sds ρ(dx) < +∞. hh, µBi 0 E

Thus we have PµB (II < +∞) = 1. On the other hand, ! X Z +∞ 1 εs = 2b(ξ )1 εs ds PµB {h(ξes)>e } PµB es {h(ξes)>e } s∈Dn 0 2 Z +∞ Z h i = ds b(ξ )1 εs ρ(dx) Pµ,x es {h(ξes)>e } hh, µBi 0 B 2 Z +∞ Z ≤ ds b(x)1{h(x)>eεs}ρ(dx) hh, µBi 0 E log+ h(x) 2 Z Z ε = b(x)ρ(dx) ds hh, µBi E 0 2 + ≤ kbk∞ log khk∞ < +∞. εhh, µBi

This implies that I is the sum of ﬁnitely many terms. Thus PµB (I < +∞) = 1. For the third term in (5.25), we have

X λ1s m e Λs h(ξes) s∈Dm

X λ1s m X λ1s m = e Λ h(ξs)1 m εs + e Λ h(ξs)1 m εs s e {Λs h(ξes)≤e } s e {Λs h(ξes)>e } s∈Dm s∈Dm =: III + IV.

In view of Deﬁnition 4.4.(iii), for III, we have

PµB (III) Z +∞ Z λ1s 2 L = e r h(ξ )1 εs Π (ξ , dr)ds PµB es {rh(ξes)≤e } es 0 E 1 Z +∞ Z Z λ1s 2 L = e ds r h(ξ )1 εs Π (ξ , dr) ρ(dx) Pµ,x es {rh(ξes)≤e } es hh, µBi 0 B E 1 Z +∞ Z Z λ1s 2 L ≤ e ds ρ(dx) r h(x)1{rh(x)≤eεs}Π (x, dr) hh, µBi 0 E E 1 Z Z Z +∞ = ρ(dx) r2h(·)ΠL(·, dr) eλ1sds hh, µBi E (0,+∞) log+ rh(·)/ε

33 Z −1 2 2 λ1/ε L = bh, r h (·)(rh(·) ∨ 1) Π (·, dr) λ1hh, µBi (0,+∞) −1 Z rh(·) = bh, ΠL(·, dr)rh(·) log∗(rh(·)) · . −λ1/ε ∗ λ1hh, µBi (0,+∞) (rh(·) ∨ 1) log (rh(·)) (5.26)

Note that the function r 7→ r is bounded from above on (0, +∞). This together (r∨1)−λ1/ε log∗ r

with (5.24) implies that the right hand side of (5.26) is ﬁnite. It follows that PµB (III < P +∞) = 1. It has been shown by (5.19) that µ m 1 m εs < +∞ = 1. This P B s∈D {Λs h(ξes)>e } implies that IV is the sum of ﬁnitely many terms. Thus we have PµB (IV < +∞) = 1. The fourth term on the right hand side of (5.25) can be dealt with similarly. In fact, we have

+∞ +∞ X λ1τi X λ1τi e Θiπ(ξeτi−, h) = e Θiπ(ξeτi−, h)1 ετi {Θiπ(ξeτi−,h)≤e } i=1 i=1 +∞ X λ1τi + e Θiπ(ξeτi−, h)1 ετi {Θiπ(ξeτi−,h)>e } i=1 =: V + VI.

Recall that given ξe (including {τi : i ≥ 1}), Θi is distributed according to η(ξeτi−, dθ) given by (4.8). Thus by Fubini’s theorem and (4.6),

PµB (V) " +∞ Z # X λ1τi = µB e π(ξeτi−, h) θ1 ετi η(ξeτi−, dθ) P {θπ(ξeτi−,h)≤e } i=1 [0,+∞) " +∞ # π(ξ , h) Z X λ1τi eτi− 2 NL = PµB e 1A(ξeτi−) θ Π (ξeτi−, dθ) ετi i=1 γ(ξeτi−) {0<θπ(ξeτi−,h)≤e } 1 Z Z +∞ π(ξ , h) λ1s es = ρ(dx)Pµ,x q(ξes)e 1A(ξes)ds hh, µBi B 0 γ(ξes) Z 2 NL · θ Π (ξes, dθ) εs {0<θπ(ξes,h)≤e } 1 Z +∞ Z π(ξ , h)2 λ1s es = e ds ρ(dx)Pµ,x 1A(ξes) hh, µBi 0 B h(ξes) Z 2 NL · θ Π (ξes, dθ) εs {0<θπ(ξes,h)≤e } 1 Z +∞ Z Z λ1s 2 2 NL ≤ e ds 1A(x)π(x, h) bh(x)m(dx) θ Π (x, dθ) hh, µBi 0 E {0<θπ(x,h)≤eεs} 1 Z Z Z +∞ = 1 (x)π(x, h)2h(x)m(dx) θ2ΠNL(x, dθ) eλ1sds A b + hh, µBi log (θπ(x,h)) E (0,+∞) ε

34 Z −1 2 2 λ1/ε NL = 1A(·)π(·, h) θ (π(·, h)θ ∨ 1) Π (·, dθ), bh λ1hh, µBi (0,+∞) Z −1 NL ∗ π(·, h)θ = Π (·, dθ)1A(·)π(·, h)θ log (π(·, h)θ) , bh . −λ1/ε ∗ λ1hh, µBi (0,+∞) (π(·, h)θ ∨ 1) log (π(·, h)θ)

Since θ 7→ θ is bounded from above on (0, +∞), we get (V) < +∞ by (5.24), (θ∨1)−λ1/ε log∗ θ PµB and hence PµB (V < +∞) = 1. We have shown in (5.20) that

+∞ ! X µB 1 ετi < +∞ = 1. P {Θiπ(ξeτi−,h)>e } i=1

Thus VI is the sum of ﬁnitely many terms and PµB (VI < +∞) = 1. The above arguments h show that the right hand side of (5.25) is ﬁnite a.s., and hence lim supt→+∞ PµB Wt (Γ) | G < h +∞ PµB -a.s. By Fatou’s lemma, PµB lim inft→+∞ Wt (Γ) | G < +∞ PµB -a.s. Let h An := µB lim inf Wt (Γ) | G ≤ n ∈ G for n ≥ 1. P t→+∞

+∞ Then PµB (∪n=1An) = 1. Since Z Z h h lim inf Wt (Γ)d µB = µB lim inf Wt (Γ) | G d µB ≤ n, t→+∞ P P t→+∞ P An An

h we get lim inft→+∞ Wt (Γ) < +∞ PµB -a.s on An for all n ≥ 1. Thus h µB lim inf Wt (Γ) < +∞ = 1. P t→+∞

h −1 Note that by [26, Proposition 2] Wt (Γ) is a non-negative PµB -supermartingale, which h −1 implies that limt→+∞ Wt (Γ) exists PµB -a.s. It follows that h PµB lim sup Wt (Γ) < +∞ = 1. t→+∞

h h This proves (5.23) and consequently PµB W∞(X) = hh, µBi. Recall that PµB W∞(X) = −1 h hgh, µBi where g(x) = h(x) Pδx W∞(X) . Thus we have

hgh, µBi = hh, µBi. (5.27)

Note that 0 ≤ g(x) ≤ 1 for every x ∈ E. We get by (5.27) that g(x) = 1 m-a.e. on B. Since

B is arbitrary, g(x) = 1 m-a.e. on E. It then follows from (5.14) that g(x) = Se tg(x) = R p(t, x, y)g(y)ρ(dy) = 1 for every x ∈ E. Therefore by (5.13), P W h (X) = hh, µi holds E e µ ∞ for all µ ∈ M(E). This completes the proof for Theorem 5.1.(i).

h h (ii) Suppose λ1 > 0. Clearly Pµ W∞(X) = 0 = 1 if and only if Pµ W∞(X) = 0. By (5.13), this would follow if g(x) = 0 for every x ∈ E. Recall that g(x) = Se tg(x) =

35 R p(t, x, y)g(y)ρ(dy). It suﬃces to prove that g(x) = 0 for m-almost every x ∈ E, or E e equivalently, h Pδx W∞(X) = 0 for m-a.e. x ∈ E. (5.28) By the change of measure methodology (see, for example, [18, Theorem 5.3.3]), (5.28) would follow if h Pδx (lim sup Wt (Γ) = +∞) = 1 for m-a.e. x ∈ E. (5.29) t→+∞

By the deﬁnition of Γt, we have

h λ1s λ1s m m Ws (Γ) = e hh, Γsi ≥ e Λs h(ξes) for s ∈ D , and W h(Γ) ≥ eλ1τi Θ π(ξ , h) for i ≥ 1. τi i eτi−

Thus under Pδx ,

h λ1s m lim sup W (Γ) ≥ lim sup e Λ h(ξ )1 m t s es {Λ h(ξes)≥1} t→+∞ Dm3s→+∞ s ∨ lim sup eλ1τi Θ π(ξ , h)1 . (5.30) i eτi− {Θiπ(ξeτ −,h)≥1} i→+∞ i

m Lemma 5.5 implies that for m-a.e. x ∈ E, both Λs h(ξes)1 m and Θiπ(ξeτi−, h)1 {Λs h(ξes)≥1} {Θiπ(ξeτi−,h)≥1} grow subexponentially. Thus when λ1 > 0, the right hand side of (5.30) goes to inﬁnity. Hence we get (5.29) for m-a.e. x ∈ E.

6 Necessary conditions for a non-degenerate martingale limit

In this section we will give necessary conditions for the fundamental martingale to have a non-degenerate limit. Recall from Proposition 4.1 that pe(t, x, y) is the transition density of the spine ξe with respect to the measure ρ. We start with the following assumption. Assumption 4.

lim sup essupy∈E|pe(t, x, y) − 1| = 0. t→+∞ x∈E

Proposition 6.1. Suppose that Assumptions 0-4 hold. Then ρ is an ergodic measure for

(Se t)t≥0, that is, ρ is an invariant probability measure for (Se t)t≥0 and for any invariant set B, either ρ(B) = 0 or ρ(B) = 1.

36 Proof. Recall from Proposition 5.3 that ρ is an invariant probability measure for (Se t)t≥0. By [16, Theorem 3.2.4], it suﬃces to prove that for any ϕ ∈ L2(E, ρ),

Z t 1 2 lim Se sϕds = hϕ, ρi in L (E, ρ). (6.1) t→+∞ t 0

It follows from Assumption 4 that for any ε > 0, there is t0 > 0 such that

sup essupy∈E|pe(s, x, y) − 1| ≤ ε for all s ≥ t0. (6.2) x∈E

For x ∈ E and t > t0,

Z t Z t0 1 1 t0 Se sϕds − hϕ, ρi = Se sϕds − hϕ, ρi t 0 t 0 t 1 Z t Z + ds (pe(s, x, y) − 1) ϕ(y)ρ(dy). (6.3) t t0 E By (6.2) and Jensen’s inequality, we have

Z t Z 1 2 k ds (pe(s, x, y) − 1) ϕ(y)ρ(dy)kL2(E,ρ) t t0 E 1 Z Z t Z 2 = ρ(dx) ds (p(s, x, y) − 1) ϕ(y)ρ(dy) 2 e t E t0 E t − t Z Z t Z ≤ 0 ρ(dx) ds (p(s, x, y) − 1)2 ϕ(y)2ρ(dy) 2 e t E t0 E 2 (t − t0) 2 2 ≤ ε kϕk 2 . (6.4) t2 L (E,ρ) Moreover, by Jensen’s inequality and (5.12),

2 Z t0 Z Z t0 1 2 1 S 2 S k e sϕdskL (E,ρ) = 2 ρ(dx) e sϕ(x)ds t 0 t E 0 t Z Z t0 0 S 2 ≤ 2 ρ(dx) e s(ϕ )(x)ds t E 0 2 Z 2 t0 2 t0 2 = 2 ϕ(x) ρ(dx) = 2 kϕkL2(E,ρ). (6.5) t E t By (6.3)–(6.5), we have

1 Z t k Se sϕds − hϕ, ρikL2(E,ρ) t 0 t0 t0 t − t0 ≤ kϕk 2 + |hϕ, ρi| + εkϕk 2 t L (E,ρ) t t L (E,ρ) 2t0 t − t0 ≤ kϕk 2 + εkϕk 2 . t L (E,ρ) t L (E,ρ)

37 Letting t → +∞ and then ε → 0, we get (6.1). Deﬁne

L E1 := {x ∈ E : supp Π (x, ·) ⊇ [N, +∞) for some N ≥ 0}, (6.6) NL E2 := {x ∈ A : supp Π (x, ·) ⊇ [N, +∞) for some N ≥ 0}. (6.7)

The main result of this section is the following theorem.

h Theorem 6.2. Suppose that Assumptions 0-4 hold. Then Pµ W∞(X) = 0 = 1 for all µ ∈ M(E) if either (5.1) fails or the following conditions hold.

λ1 ≥ 0 and m(E1 ∪ E2) > 0. To prove Theorem 6.2, we need the following lemma. Lemma 6.3. Suppose that Assumptions 0-4 hold.

(i) If m(E1 ∪ E2) > 0, then for m-almost every x ∈ E,

m lim sup Λs h(ξes) ∨ lim sup Θiπ(ξeτi−, h) = +∞ P·, x-a.s.; (6.8) Dm3s→+∞ i→+∞

(ii) if (5.1) fails, then for m-almost every x ∈ E,

+ m + log Λs h(ξes) log Θiπ(ξeτi−, h) lim sup ∨ lim sup = +∞ P·, x-a.s. (6.9) Dm3s→+∞ s i→+∞ τi

Proof. It is easy to see that (6.9) is equivalent to saying that for m-almost every x ∈ E and all λ < 0,

λs m λτi lim sup e Λs h(ξes) ∨ lim sup e Θiπ(ξeτi−, h) = +∞ P·, x-a.s. Dm3s→+∞ i→+∞ We divide the conditions of this lemma into two cases, and prove the results separately. Case I: Suppose either one of the following conditions holds:

(I.a) m(E1) > 0; Z (I.b) ( rh(·) log+(rh(·))ΠL(·, dr), bh) = +∞. (0,+∞) Let λ < 0 be an arbitrary constant. To prove (6.8) ( resp. (6.9)) under condition (I.a) (resp. (I.b)), it suﬃces to prove that for m-a.e. x ∈ E and any M ≥ 1, ! X ·, x 1 m = +∞ = 1 P {Λs h(ξes)≥M} s∈Dm

38 ! X (resp. ·, x 1 λs m = +∞ = 1 ). (6.10) P {e Λs h(ξes)≥M} s∈Dm For 0 ≤ s ≤ t < +∞, θ ≤ 0 and M ≥ 1, let

Z t Z L I (s, t) := dr u1 θr Π (ξ , du), θ {e uh(ξer)≥M} er s (0,+∞)

m θs m and Iθ(t) := Iθ(0, t). Recall that, given ξe, for any T > 0, #{s ∈ DT : e Λs h(ξes) ≥ M}

is a Poisson random variable with parameter Iθ(T ). Hence (6.10) would follow if for m-a.e. x ∈ E,

P·, x (I0(∞) = +∞) = 1 (resp. P·, x (Iλ(∞) = +∞) = 1) (6.11) R under condition (I.a) (resp. (I.b)). Let ν(dx) := bh(x)m(dx). Clearly P·,ν = E P·, x ρ(dx). Recall that ρ is an invariant measure for Se t. By Fubini’s theorem, Z Z T Z L (I (T )) = dr u1 θr Π (ξ , du) ρ(dx) P·,ν θ P·, x {e uh(ξer)≥M} er E 0 (0,+∞) Z T Z Z L = dr u1 θr Π (ξ , du) ρ(dx) P·, x {e uh(ξer)≥M} er 0 E (0,+∞) Z T Z Z L = dr ρ(dx) u1{eθruh(x)≥M}Π (x, du). (6.12) 0 E (0,+∞)

R 2 L By the boundedness of h and x 7→ (0,+∞)(u ∧ u )Π (x, du), we have

P·,ν (Iθ(T )) Z Z ≤ T bh(x)m(dx) h(x)uΠL(x, du) E u≥M/h(x) Z Z 1 = T bh(x)m(dx) h(x)(1 ∨ )(u ∧ u2)ΠL(x, du) E u≥M/h(x) u Z Z khk∞ 2 L ≤ T 1 ∨ k (u ∧ u )Π (x, du)k∞ h(x)bh(x)m(dx) < +∞. M (0,+∞) E

Thus P·,ν(Iθ(T ) < +∞) = 1. On the other hand, by the Markov property of ξe and (6.12),

2 P·,ν Iθ(T ) Z 2 = P·, x Iθ(T ) ρ(dx) E Z Z T Z L = 2 ρ(dx) ds u1 θs Π (ξ , du) P·, x {e uh(ξes)≥M} es E 0 (0,+∞) Z T Z L · dr v1 θr Π (ξ , dv) {e vh(ξer)≥M} er s (0,+∞)

39 Z Z T Z L = 2 ρ(dx) ds u1 θs Π (ξ , du) P·, x {e uh(ξes)≥M} es E 0 (0,+∞) Z T −s Z L · dr v1 θ(r+s) Π (ξ , dv) P·,ξes {e vh(ξer)≥M} er 0 (0,+∞) Z Z T Z L = 2 ρ(dx) ds u1{eθsuh(x)≥M}Π (x, du) E 0 (0,+∞) Z T −s Z L · dr v1 θ(r+s) Π (ξ , dv) P·, x {e vh(ξer)≥M} er 0 (0,+∞) Z Z T Z L ≤ 2 ρ(dx) ds u1{eθsuh(x)≥M}Π (x, du) E 0 (0,+∞) Z T Z L · dr v1 θr Π (ξ , dv) P·, x {e vh(ξer)≥M} er 0 (0,+∞) Z Z T Z L = 2 ρ(dx) ds u1{eθsuh(x)≥M}Π (x, du)P·, x (Iθ(T )) . (6.13) E 0 (0,+∞)

Assumption 4 implies that there are constants t1, δ > 0 such that

sup essupy∈Epe(t, x, y) ≤ 1 + δ for all t ≥ t1. (6.14) x∈E

Using Fubini’s theorem, (6.14) and (6.12), we have for T > t1,

Z T Z Z L P·, x [Iθ(t1,T )] = dr pe(r, x, y)ρ(dy) v1{eθrvh(y)≥M}Π (y, dv) t1 E (0,+∞) Z T Z Z L ≤ (1 + δ) dr ρ(dy) v1{eθrvh(y)≥M}Π (y, dv) t1 E (0,+∞)

≤ (1 + δ)P·,ν (Iθ(T )) . (6.15)

On the other hand, for x ∈ E,

P·, x (Iθ(t1)) Z t1 Z L = dr v1 θr Π (ξ , dv) P·, x {e vh(ξer)≥M} er 0 (0,+∞) Z t1 Z Z L = dr pe(r, x, y)ρ(dy) v1{eθrvh(y)≥M}Π (y, dv) 0 E (0,+∞) Z t1 Z Z 1 2 L ≤ dr pe(r, x, y)ρ(dy) 1 ∨ v ∧ v Π (y, dv) 0 E v≥M/h(y) v

Z khk∞ 2 L ≤ t1 1 ∨ k (v ∧ v )Π (·, dv)k∞ =: c1 < +∞. (6.16) M (0,+∞)

40 It follows from (6.15) and (6.16) that for T > t1,

P·, x (Iθ(T )) = P·, x (Iθ(t1)) + P·, x (Iθ(t1,T )) ≤ c1 + (1 + δ)P·,ν (Iθ(T )) .

This together with (6.12) and (6.13) implies that

2 2 P·,ν Iθ(T ) ≤ 2c1P·,ν (Iθ(T )) + 2(1 + δ)P·,ν (Iθ(T )) .

Hence by the Cauchy-Schwarz inequality, we have

2 1 P·,ν(Iθ(T )) P·,ν(Iθ(T )) P·,ν Iθ(T ) ≥ P·,ν(Iθ(T )) ≥ 2 ≥ . (6.17) 2 4P·,ν(Iθ(T ) ) 8c1 + 8(1 + δ)P·,ν(Iθ(T ))

R R L Note that P·,ν(I0(T )) = T E ρ(dx) u≥h(x)/M uΠ (x, du). Condition (I.a) implies that the integral on the right hand side is positive. Hence P·,ν(I0(T )) → +∞ as T → +∞. On the other hand, note that by (6.12) and Fubini’s theorem, for λ < 0,

P·,ν (Iλ(T )) Z Z Z T L ≤ h(x)m(dx) h(x)uΠ (x, du) 1 + ds b {s≤ log (h(x)u)−log M } E (0,+∞) 0 −λ Z Z log+(h(x)u) − log M + = bh(x)m(dx) h(x)u T ∧ ΠL(x, du). E (0,+∞) −λ

Clearly condition (I.b) implies that limT →+∞ P·,ν(Iλ(T )) = +∞. Thus by letting T → +∞ in (6.17), we get P·,ν(I0(∞) = +∞) > 0 (resp. P·,ν(Iλ(∞) = +∞) > 0) under condition (I.a) (resp. (I.b)). Since {I0(∞) = +∞} (resp. {Iλ(∞) = +∞}) is an invariant event of the canonical dynamic system associated with (Se t)t≥0 and ergodic measure ρ, it follows from [16, Theorem 1.2.4] that P·,ν(I0(∞) = +∞) = 1 (resp. P·,ν(Iλ(∞) = +∞) = 1) under condition (I.a) (resp. (I.b)). Hence we prove (6.11). Case II. Suppose either one of the following conditions holds:

(II.a) m(E2) > 0; Z (II.b) ( π(·, h)r log+(π(·, h)r)ΠNL(·, dr), bh) = +∞. (0,+∞) Let λ < 0 be an arbitrary constant. To prove (6.8) ( resp. (6.9)) under condition (II.a) (resp. (II.b)), it suﬃces to prove that for m-a.e. x ∈ E and any M ≥ 1,

+∞ ! X ·, x 1 = +∞ = 1 P {Θiπ(ξeτi−,h)≥M} i=1 +∞ ! X (resp. ·, x 1 λτi = +∞ = 1 ). (6.18) P {e Θiπ(ξeτi−,h)≥M} i=1

41 The main idea of this proof is similar to that of Case I. For all T > 0, θ ≤ 0 and M ≥ 1, let P IIθ(T ) := 1 θτi . For any s ≥ 0 and x ∈ E, deﬁne τi≤T {e Θiπ(ξeτi−,h)≥M} Z fθ(s, x) := 1{eθsuπ(x,h)≥M}η(x, du) [0,+∞) Z 1 NL = 1A(x) 1{eθsuπ(x,h)≥M}uΠ (x, du), γ(x) (0,+∞)

gθ(s, x) := q(x)fθ(s, x) Z π(x, h) NL = 1A(x) 1{eθsuπ(x,h)≥M}uΠ (x, du). h(x) (0,+∞)

Recall from Deﬁnition 4.4 that, given ξe (including {τi : i ≥ 1}), Θi is distributed according

to η(ξeτi−, dr). By (4.6), we have for x ∈ E, " # X Z (II (T )) = 1 θτ η(ξ , dr) P·, x θ P·, x {e i rπ(ξeτ −,h)≥M} eτi− [0,+∞) i τi≤T " # X Z T = P·, x fθ(τi, ξeτi−) = P·, x q(ξes)fθ(s, ξes)ds 0 τi≤T Z T = P·, x gθ(s, ξes)ds . (6.19) 0

We still use ν to denote the measure bh(x)m(dx). Since ρ is an invariant measure for Se t, by Fubini’s theorem, Z P·,ν (IIθ(T )) = P·, x (IIθ(T )) ρ(dx) E Z T Z = ds P·, x gθ(s, ξes) ρ(dx) 0 E Z T Z = ds gθ(s, x)ρ(dx) (6.20) 0 E Z Z Z T NL = bh(x)m(dx) π(x, h)rΠ (x, dr) 1{eθsrπ(x,h)≥M}ds. (6.21) A (0,+∞) 0 It then follows by Assumption 3.(ii) that Z Z NL RHS of (6.21) ≤ T k yΠ (·, dy)k∞ π(x, h)bh(x)m(dx) < +∞. (0,+∞) A

Therefore P·,ν (IIθ(T ) < +∞) = 1. Recall that given ξe (including {τi : i ≥ 1}), {Θi : i ≥ 1} are mutually independent, we have

2 P·, x IIθ(T ) − P·, x (IIθ(T ))

42   X = P·, x  1 θτi 1 θτj  {e Θiπ(ξeτi−,h)≥M} {e Θj π(ξeτj −,h)≥M} τi,τj ≤T,i6=j   X Z Z = ·, x 1 θτ η(ξτ −, dy) 1 θτj η(ξτ −, dz) P  {e i yπ(ξeτ −,h)≥M} e i {e zπ(ξeτ −,h)≥M} e j  [0,+∞) i [0,+∞) j τi,τj ≤T,i6=j   X = P·, x  fθ(τi, ξeτi−)fθ(τj, ξeτj −) . τi,τj ≤T,i6=j

Thus by (4.7),

2 P·, x IIθ(T ) − P·, x (IIθ(T )) Z T Z Z T −s h = 2P·, x q(ξes)fθ(s, ξes)ds Πe y q(ξer)fθ(s + r, ξer)dr π (ξes, dy) 0 E 0 Z T Z Z T −s h = 2P·, x gθ(s, ξes)ds Πe y gθ(s + r, ξer)dr π (ξes, dy) . (6.22) 0 E 0

Note that for all x ∈ E and θ ≤ 0, s 7→ gθ(s, x) is non-increasing. Thus it follows from (6.22) that Z T Z Z T 2 h P·, x IIθ(T ) ≤ 2P·, x gθ(s, ξes)ds Πe y gθ(r, ξer)dr π (ξes, dy) 0 E 0

+ P·, x (IIθ(T )) . (6.23)

By Fubini’s theorem, (6.14) and (6.20), we have for y ∈ E and T > t1,

Z T Z T Z Πe y gθ(s, ξes)ds = ds pe(s, y, z)gθ(s, z)ρ(dz) t1 t1 E Z T Z ≤ (1 + δ) ds gθ(s, z)ρ(dz) t1 E

≤ (1 + δ)P·,ν (IIθ(T )) . (6.24)

On the other hand, by Assumption 3.(iii),

Z t1 sup Πe y gθ(s, ξes)ds y∈E 0 Z t1 Z Z π(z, h) NL = sup ds pe(s, y, z) ρ(dz) r1{eθsrπ(z,h)≥M}Π (z, dr) y∈E 0 A h(z) (0,+∞) Z Z t1 Z π(·, h) NL ≤ k 1Ak∞k rΠ (·, dr)k∞ sup ds pe(s, y, z)ρ(dz) h (0,+∞) y∈E 0 A

≤ c2t1 =: c3 < +∞.

43 This and (6.24) imply that

Z T Πe y gθ(s, ξes)ds ≤ c3 + (1 + δ)P·,ν(IIθ(T )) for all y ∈ E and T > t1. 0 This together with (6.23) and (6.19) implies that

2 P·, x IIθ(T ) ≤ (1 + 2c3)P·, x (IIθ(T )) + 2(1 + δ)P·,ν (IIθ(T )) P·, x (IIθ(T )) .

Consequently, Z 2 2 P·,ν IIθ(T ) = P·, x IIθ(T ) ρ(dx) E 2 ≤ (1 + 2c3)P·,ν (IIθ(T )) + 2(1 + δ)P·,ν (IIθ(T )) .

R R NL Recall that P·,ν(II0(T )) = T A π(x, h)bh(x)m(dx) r≥M/π(x,h) rΠ (x, dr). Condition (II.a) implies that the integral on the right hand side is positive. Thus P·,ν(II0(T )) → +∞ as T → +∞. On the other hand, note that by (6.21) and Fubini’s theorem, for λ < 0,

Z Z + + NL log (π(x, h)r) − log M P·,ν (IIλ(T )) = bh(x)m(dx) Π (x, dr)π(x, h)r ∧ T . A (0,+∞) −λ

Clearly condition (II.b) implies that limT →+∞ P·,ν(IIλ(T )) = +∞. Similarly by using the Cauchy-Schwarz inequality and letting T → +∞, we get P·,ν(II0(∞) = +∞) > 0 (resp. P·,ν(IIλ(∞) = +∞) > 0) under condition (II.a) (resp. (II.b)). For each n ≥ 1, we denote by Gen the σ-ﬁeld generated by ξe up to time τn (including

{τ1, ··· , τn}) and {Θi : i ≤ n}. Obviously for each i ≥ 1, both Z 1 θτ and 1 θτ η(ξ , dr) {e i Θiπ(ξeτ −,h)≥M} {e i rπ(ξeτ −,h)≥M} eτi− i [0,+∞) i are Gei-measurable. Moreover, for every x ∈ E, under P·, x, ·, x 1 θτi+1 | Gei P {e Θi+1π(ξeτi+1−,h)≥M} Z = 1 θτ η(ξ , dr) | G . P·, x {e i+1 rπ(ξeτ −,h)≥M} eτi+1− ei [0,+∞) i+1 Applying the second Borel-Cantelli lemma (see, for example, [18, Corollary 5.3.2]) to both sides of the above equality, we get that

( +∞ ) ( +∞ ) X X Z 1 θτi = +∞ = 1 θτi η(ξeτi−, dr) = +∞ {e Θiπ(ξeτi−,h)≥M} {e rπ(ξeτi−,h)≥M} i=1 i=1 [0,+∞) under P·, x. It is easy to see from the above representation that {II0(∞) = +∞} (resp. {IIλ(∞) = +∞}) is an invariant event of the canonical dynamic system associated with

44 (Se t)t≥0 and ergodic measure ρ, so it follows from [16, Theorem 1.2.4] that P·,ν(II0(∞) = +∞) = 1 (resp. P·,ν(IIλ(∞) = +∞) = 1) under condition (II.a) (resp. (II.b)). Thus (6.18) is valid. Proof of Theorem 6.2. Applying the same argument as in the beginning of the proof of Theorem 5.1.(ii) here, we only need to show that under the assumptions of Theorem 6.2,

h Pδx (lim sup Wt (Γ) = +∞) = 1 for m-a.e. x ∈ E. t→+∞

In view of (5.30), this would follow if for m-a.e. x ∈ E,

λ1s m λ1τi lim sup e Λs h(ξes) ∨ lim sup e Θiπ(ξeτi−, h) = +∞ P·, x-a.s. Dm3s→+∞ i→+∞

which, under the assumptions of this theorem, is automatically true by Lemma 6.3. Hence we complete the proof. The following corollaries follow directly from Theorem 5.1 and Theorem 6.2.

Corollary 6.4. Suppose that Assumptions 0-4 hold and that m(E1 ∪ E2) > 0 with E1 and 0 h E2 deﬁned in (6.6) and (6.7) respectively. For every µ ∈ M(E) , W∞(X) is non-degenerate

if and only if λ1 < 0 and condition (5.1) holds. Moreover, Xt under Pµ exhibits weak local

extinction if λ1 ≥ 0.

0 h Corollary 6.5. Suppose Assumptions 0-4 hold and λ1 < 0. For every µ ∈ M(E) , W∞(X) is non-degenerate if and only if condition (5.1) holds. Remark 6.6. Note that in the case of purely local branching mechanism, Assumption 4 can be written as h lim sup essupy∈E p (t, x, y) − 1 = 0, t→+∞ x∈E where ph(t, x, y) denotes the transition density function of ξh with respect to the measure ρ. If E is a bounded domain in Rd, m is the Lebesgue measure on Rd and ξ is a symmetric 2 diﬀusion on E, then a(x) ∈ Bb(E) ⊂ K(ξ) ∩ L (E, m). Hence for the class of superdiﬀusions with local branching mechanisms considered in [37], our Assumptions 0-4 hold naturally and Corollary 6.5 generalizes [37, Theorem 1.1].

7 Examples

In this section, we will give examples satisfying Assumptions 0-4. We will not try to give the most general examples possible.

45 Example 7.1. Suppose E = {1, 2, ··· ,K} (K ≥ 2), m is the counting measure on E and + Stf(i) = f(i) for all i ∈ E, t ≥ 0 and f ∈ B (E) (that is, there is no spatial motion). Suppose Z φL(i, λ) := a(i)λ + b(i)λ2 + e−λr − 1 + λr ΠL(i, dr), (0,+∞) Z φNL(i, f) := −c(i)π(i, f) − 1 − e−rπ(i,f) ΠNL(i, dr), (0,+∞) where for each i ∈ E, a(i) ∈ (−∞, +∞), b(i), c(i) ≥ 0, (r ∧ r2)ΠL(i, dr) and rΠNL(i, dr) are R NL bounded kernels from E to (0, +∞) with {i ∈ E : (0,+∞) rΠ (i, dr) > 0}= 6 ∅, and π(i, dj) is a probability kernel on E with π(i, {i}) = 0 for every i ∈ E. As a special case of the model

given in Section 2.1, we have a non-local branching superprocess {Xt : t ≥ 0} in M(E) with transition probabilities given by

+ Pµ [exp (−hf, Xti)] = exp (−hVtf, µi) for µ ∈ M(E), t ≥ 0 and f ∈ Bb (E),

where Vtf(i) is the unique non-negative locally bounded solution to the following integral equation:

Z t L NL Vtf(i) = f(i) − φ (i, Vsf(i)) + φ (i, Vsf) ds for t ≥ 0, i ∈ E. 0

For every i ∈ E and µ ∈ M(E), we define µ(i) := µ({i}). The map µ 7→ (µ(1), ··· , µ(K))T is clearly a homeomorphism between M(E) and the K-dimensional product space [0, +∞)K . (1) (K) T K Hence {(Xt , ··· ,Xt ) : t ≥ 0} is a Markov process in [0, +∞) , which is called a K-type continuous-state branching process. (Clearly the 1-type continuous-state branching process defined in a similar way coincides with the classical one-dimensional continuous-state branching process, see, for example, [36, Chapter 3].) For simplicity, we assume b(i) ≡ 0. R NL For i, j ∈ E, let pij := π(i, {j}) and γ(i) := c(i) + (0,+∞) rΠ (i, dr). Define the K × K h (j)i matrix M(t) = (M(t)ij)ij by M(t)ij := Pδi Xt for i, j ∈ E. Let Pt denote the mean semigroup of X, that is

K X + Ptf(i) := Pδi [hf, Xti] = M(t)ijf(j) for i ∈ E, t ≥ 0 and f ∈ B (E). j=1

By the Markov property and (2.9), M(t) satisﬁes that

M(0) = I, M(t + s) = M(t)M(s) for t, s ≥ 0,

K Z t X Z t and M(t)ij = δj(i) − a(i) M(s)ijds + γ(i) pik M(s)kjds 0 k=1 0

46 for i, j ∈ E. This implies that M(t) has a formal matrix generator A := (Aij)ij given by

+∞ X An M(t) = eAt = tn , and A = γ(i)p − a(i)δ (j) for i, j ∈ E. n! ij ij i n=0

We assume A is an irreducible matrix. It then follows by [3, Lemma A.1] that M(t)ij > 0 for

all t > 0 and i, j ∈ E. Let Λ := supλ∈σ(A) Re(λ) where σ(A) denotes the set of eigenvalues of A. The Perron-Frobenius theory (see, for example, [3, Lemma A.3]) tells us that for every t > 0, eΛt is a simple eigenvalue of M(t), and there exist a unique positive right eigenvector T T u = (u1, ··· , uK ) and a unique positive left eigenvector v = (v1, ··· , vK ) such that

K K X X Λt T Λt ui = uivi = 1, M(t)u = e u, v M(t) = e v. i=1 i=1 Moreover it is known by [3, Lemma A.3] that for each i, j ∈ E,

−Λt e M(t)ij → uivj as t → +∞. (7.1)

−1 One can easily verify that Assumptions 0-3 hold with λ1 = −Λ, h(i) = cui and bh(i) = c vi, −1/2 PK 2 h −Λt PK (i) where c := j=1 uj is a positive constant. Thus Wt (X) := ce i=1 uiXt is a non-negative martingale. Applying Theorem 4.6 here, we can deduce that under the martingale change of measure the spine process ξe is a continuous-time Markov process on E

with intensity matrix Q = (qij)ij given by PK γ(i) j=1 pijuj γ(i)pijuj qii := − = −(Λ + a(i)), qij := for i 6= j. ui ui PK Let ρ(dj) := ujvjm(dj) = i=1 ujvjδi(dj). Let Se t denote the transition semigroup of the spine ξe and pe(t, i, j) denote its transition density with respect to ρ. It follows by Proposition 4.1 that for each i, j ∈ E, Z pe(t, i, j)ujvj = pe(t, i, k)δj(k)ρ(dk) = Se tδj(i) E e−Λt e−Λt = Pt(hδj)(i) = M(t)ijuj. (7.2) h(i) ui

−Λt −1 Thus pe(t, i, j) = e (uivj) M(t)ij. By (7.1), we have for each i, j ∈ E.

pe(t, i, j) → 1 as t → +∞. Hence Assumption 4 also holds for this example. Applying Corollary 6.4 here, we conclude that for every non-trivial µ ∈ M(E), the martingale limit

K h h −Λt X (i) W∞(X) := lim Wt (X) = lim ce uiXt t→+∞ t→+∞ i=1

47 is non-degenerate if and only if Λ > 0 and

K Z X + L uivi r log (rui)Π (i, dr) i=1 (0,+∞) K K Z K X X + X NL + pijujvi r log (r pikuk)Π (i, dr) < +∞. i=1 j=1 (0,+∞) k=1 Using elementary computation , one can reduce the above condition to Z Z r log+ rΠL(i, dr) + r log+ rΠNL(i, dr) < +∞ ∀i ∈ E. (7.3) (0,+∞) (0,+∞)

(i) In particular, under condition (7.3), Pµ limt→+∞ Xt = 0 = 1 for every i ∈ E and every non-trivial µ ∈ M(E) if and only if Λ ≤ 0. This result is also proved in [35, Theorem 6]. Now we give some other examples.

Example 7.2. Suppose that E is a bounded C3 domain in Rd (d ≥ 1) , m is the Lebesgue L measure on E and that ξ = (ξt, Πx) is the killed Brownian motion in E. Suppose that φ and φNL are as given in Subsection 2.1. We assume Assumption 0 holds. We further assume that the probability kernel π(x, dy) has a bounded density with respect to the Lebesgue measure m, i.e., π(x, dy) = π(x, y)dy with π(x, y) being bounded on E × E. Assumption 1 and

Assumption 3.(i1) are trivially satisﬁed. Let (Pt)t≥0 be the semigroup on Bb(E) uniquely determined by the integral equation (2.9). It follows from [27, Theorem] that Assumption 2,

Assumption 3.(ii) are satisﬁed, and that (Pt)t≥0 is uniformly primitive in the sense of [27]. + Thus for all t > 0, f ∈ Bb (E) and x ∈ E,

−λ1t −λ1t Ptf(x) − e (f, bh)h(x) ≤ cte (f, bh)h(x), (7.4)

where ct ≥ 0 satisfying ct ↓ 0 as t ↑ +∞, λ1 is the constant in Assumption 2, and h, bh are the λ1t −1 + functions in Assumption 2. Let Se tf(x) := e h(x) Pt(fh)(x) for f ∈ B (E), t ≥ 0 and x ∈ E. Let pe(t, x, y) be the density of SgP t with respect to the measure ρ(dy) := h(y)bh(y)dy + on E. By (7.4), we have for every t > 0, f ∈ Bb (E) and x ∈ E, Z

Se tf(x) − hf, ρi = (pe(t, x, y) − 1) f(y)ρ(dy) ≤ cthf, ρi. E It follows from this that

sup essupy∈E |pe(t, x, y) − 1| ≤ ct → 0 as t → +∞. x∈E Hence Assumption 4 is satisfied. Assumption 3.(iii) will be satisfied if the function π(x, y) satisfies Z π(x, y)h(y)dy ≤ ch(x) ∀x ∈ {z ∈ E : γ(z) > 0} E

48 for some constant c > 0, where h is the function in Assumption 2 and γ(z) is as given in Subsection 2.1.

Example 7.3. Suppose that E is a bounded C1,1 open set in Rd (d ≥ 1), m is the Lebesgue measure on E, α ∈ (0, 2), β ∈ [0, α∧d) and that ξ = (ξt, Πx) is an m-symmetric Hunt process on E satisfying the following conditions: (1) ξ has a L´evysystem (N, t) where N = N(x, dy) is a kernel given by C N(x, dy) = 1 dy x, y ∈ E |x − y|d+α for some constant C1 > 0. (2) ξ admits a jointly continuous transition density p(t, x, y) with respect to the Lebesgue measure and that there exists a constant C2 > 1 such that

−1 C2 qα,β(t, x, y) ≤ p(t, x, y) ≤ C2qα,β(t, x, y) ∀(t, x, y) ∈ (0, 1] × E × E, where δ (x)β δ (y)β t q (t, x, y) = 1 ∧ E 1 ∧ E t−d/α ∧ . (7.5) α,β t1/α t1/α |x − y|d+α

Here δE(x) stands for the Euclidean distance between x and the boundary of E. Suppose that φL and φNL are as given in Subsection 2.1. We assume Assumption 0 holds. We further assume that the probability kernel π(x, dy) has a density π(x, y) with respect to the Lebesgue measure m satisfying the condition

−d π(x, y) ≤ C3|x − y| ∀x, y ∈ E for some positive constants C3 and . Deﬁne

−1 d+α F (x, y) := C1 |x − y| γ(x)π(x, y) ∀x, y ∈ E.

One can show easily that Assumption 1 and Assumption 3.(i2) are satisﬁed. Deﬁne

F ∗(x, y) := log (1 + F (x, y)) ∀x, y ∈ E.

It is obvious that there exists C4 > 0 such that

∗ +α 0 ≤ F (x, y) ≤ C4 |x − y| ∧ 1 ∀x, y ∈ E,

∗ and thus, by [10, Proposition 4.2], F belongs to the Kato class Jα,β deﬁned in [10], i.e., α,β limt↓0 NF ∗ (t) = 0, where

Z t Z 1/α β ∗ ∗ α,β |z − y| ∧ t F (y, z) + F (z, y) NF ∗ (t) := sup qα,β(s, x, z) 1 + d+α dydzds. x∈E 0 E×E |x − z| |z − y|

49 The measure µ(dx) := −a(x)dx obviously belongs to the Kato class Kα,β deﬁned in [10], α,β i.e., limt↓0 Nµ (t) = 0, where Z t Z α,β Nµ (t) = sup qα,β(s, x, y)|a|(y)dyds, x∈E 0 E R t P ∗ since a is a bounded function. For 0 ≤ t < +∞, let At := − 0 a(ξr)dr + 0

+ Ttf(x) := Πx [exp (At) f(ξt)] , t ≥ 0, x ∈ E, f ∈ B (E).

Now it follows from [10, Theorem 1.3] that the semigroup (Tt)t≥0 has a jointly continuous density q(t, x, y) with respect to the Lebesgue measure and there exists a constant C5 > 1 such that

−1 C5 qα,β(t, x, y) ≤ q(t, x, y) ≤ C5qα,β(t, x, y) ∀(t, x, y) ∈ (0, 1] × E × E. (7.6)

Let (Tbt)t≥0 be the dual semigroup of (Tt)t≥0. By (7.6), one can easily show that for any f ∈ Bb(E), Ttf and Tbtf are bounded continuous functions on E, that Tt and Tbt are bounded 2 ∞ operators from L (E, m) into L (E, m), and that (Tt)t≥0 and (Tbt)t≥0 are strongly continuous 2 semigroups on L (E, m). Let L and Lb be the generators of (Tt)t≥0 and (Tbt)t≥0 respectively. Let σ(L) and σ(Lb) denote the spectrum of L and Lb respectively. It follows from (7.6) and Jentzsch’s theorem ([39, Theorem V.6.6, p. 337]) that the common value −λ1 := sup Re(σ(L)) = sup Re(σ(Lb)) is an eigenvalue of multiplicity 1 for both L and Lb, and that an eigenfunction h of L associated with −λ1 is bounded continuous and can be chosen strictly positive on E and satisfies khkL2(E,m) = 1, and that an eigenfunction bh of Lb associated with −λ1 is bounded continuous and can be chosen strictly positive on E and satisfies (h, bh) = 1. Thus Assumption 2 and 3.(ii) are satisfied. It follows from (7.6) and the −λ1 −λ1 equations e h = T1h, e bh = Tb1h that there exists a constant C6 > 1 such that

−1 β β −1 β β C6 δE(x) ≤ h(x) ≤ C6δE(x) ,C6 δE(x) ≤ bh(x) ≤ C6δE(x) ∀x ∈ E.

It follows from this, (7.6) and the semigroup properpty that the semigroups (Tt)t≥0 and

(Tbt)t≥0 are intrinsically ultracontractive. For the definition of intrinsic ultracontractivity, see λ1t −1 + [31]. Let Se tf(x) := e h(x) Tt(fh)(x) for f ∈ B (E), t ≥ 0 and x ∈ E. Then Se t admits a density pe(t, x, y) with respect to the probability measure h(y)bh(y)dy which is related to q(t, x, y) by eλ1tq(t, x, y) pe(t, x, y) = ∀(t, x, y) ∈ (0, +∞) × E × E. h(x)bh(y) Now it follows from [31, Theorem 2.7] that Assumption 4 is satisfied. As in the previous example, Assumption 3.(iii) will be satisfied if the function π(x, y) satisfies Z π(x, y)h(y)dy ≤ ch(x) ∀x ∈ {z ∈ E : γ(z) > 0} E

50 for some constant c > 0, where γ(z) is as given in Subsection 2.1. One concrete example of ξ is the killed symmetric α-stable process in E. In this case, (7.5) is satisfied with β = α/2, a fact which was first proved in [8]. Another concrete example of ξ is the censored symmetric α-stable process in E introduced in [4] when α ∈ (1, 2). In this case, (7.5) is satisfied with β = α − 1, a fact which was first proved in [9]. In fact, by using [10], one could also include the case when E is a d-set, α ∈ (0, 2) and ξ is an α-stable-like process in E introduced in [11]. We omit the details.

Example 7.4. Suppose that E = Rd, m is the Lebesgue measure on Rd, α ∈ (0, 2) and that ξ = (ξt, Πx) is a Markov process corresponding to the Feynman-Kac transform of a d-dimensional isotropic α-stable process with killing potential η(x) = |x|β (β > 0). Let J(x) = J(|x|) be the Lévydensity of the isotropic α-stable process, i.e., J(x) = c(d, α)|x|−d−α for some positive constant c(d, α) depending only on d and α. It is known that ξ has a Lévy system (N, t) where N(x, dy) = 2J(y − x)dy. Let (E, F) be the Dirichlet form of ξ. Then E has the following form Z Z Z E(u, v) = (u(x) − u(y))(v(x) − v(y))J(y − x)dxdy + u(x)v(x)|x|βdx ∀u, v ∈ F. Rd Rd Rd Suppose that the branching mechanisms φL and φNL are as given in Subsection 2.1. For simplicity we assume a(x) ≡ 0 and Assumption 0 holds. Let π(x) = π(|x|) be a probability density on Rd such that the function π(x)/J(x) is bounded from above. We assume that the probability kernel π(x, dy) has a density π(x, y) = π(y − x) with respect to the Lebesgue measure and that the function γ(x) ≡ γ is a constant. Define γ(x, y) := γπ(y − x). Then R γ(x, y)dx = γ R π(y − x)dx = γ, and Assumption 1 is trivially satisfied. Define Rd Rd γ(x, y) γπ(y − x) F (x, y) := = ∀x, y ∈ d. 2J(y − x) 2J(y − x) R Then F is a bounded function on Rd × Rd vanishing on the diagonal, and thus Assumption 3.(i2) is satisfied. Define X Z t Z Aet := log(1 + F (ξs−, ξs)) − 2 F (ξs, y)J(y − ξs)dyds d s≤t 0 R X Z t Z = log(1 + F (ξs−, ξs)) − γ π(y − ξs)dyds d s≤t 0 R X = log(1 + F (ξs−, ξs)) − γt. s≤t It follows from [14, Theorem 4.8] (see also [15, p. 275]) that the bilinear form corresponding to the symmetric semigroup h i Aet d d Tetf(x) = Πx e f(ξt) ∀t ≥ 0, x ∈ R , f ∈ Bb(R )

51 is Z Z Qe(u, v) = (u(x) − u(y))(v(x) − v(y))(1 + F (x, y))J(y − x)dxdy d d Z R R + u(x)v(x)|x|βdx ∀u, v ∈ F. Rd This is the bilinear form of the stable-like L´evyprocess killed with the potential η(x) = |x|β.

Now we apply [29, Examples 4.5 and 4.8] to get that the symmetric semigroup (Tet)t≥0 is intrinsically ultracontractive. Deﬁne X At := log(1 + F (ξs−, ξs)). s≤t

Again by [14, Theorem 4.8] (see also [15, p. 275]) that the bilinear form corresponding to the symmetric semigroup

At d d Ttf(x) = Πx e f(ξt) ∀t ≥ 0, x ∈ R , f ∈ Bb(R ) is Z Z Q(u, v) = E(u, v) − 2 u(y)v(x)F (x, y)J(x − y)dxdy d d Z RZ R = E(u, v) − u(y)v(x)γ(x, y)dxdy ∀u, v ∈ F. Rd Rd

γt We observe that Ttf = e Tetf. So the semigroup (Tt)t≥0 is also intrinsically ultracontractive. Applying similar argument as in Example 7.3, one can show that Assumptions 2 and 4 are satisfied. Finally, Assumption 3(iii) (and thus Assumption 3(ii)) will be satisfied if the function π(x) satisfies that Z d π(y − x)h(y)dy ≤ ch(x) ∀x ∈ R Rd for some positive constant c. Here h is the strictly positive eigenfunction associated with the principal eigenvalue of the generator of the semigroup (Tt)t≥0.

Appendix

Proof of Proposition 4.3: We will prove (4.6) ﬁrst. We claim that

Z t∧τ1 Z h i h Πe x f(τ1, ξeτ1−, ξeτ1 )1{τ1≤t} = Πe x q(ξes)ds f(s, ξes, y)π (ξes, dy) . (7.7) 0 E

52 It is easy to see from the construction of ξe that Z t Z h h h h h LHS of (7.7) = Πx q(ξs )eq(s)ds f(s, ξs , y)π (ξs , dy) . (7.8) 0 E On the other hand, by Fubini’s theorem, we have

Z t Z h RHS of (7.7) = ds Πe x q(ξes)f(s, ξes, y)π (ξes, dy)1{s<τ1} 0 E Z t Z = ds Πh q(ξ )f(s, ξ , y)πh(ξ , dy)1 x bs bs bs {s<ζb} 0 E Z t Z h h h h h = ds Πx eq(s) q(ξs )f(s, ξs , y)π (ξs , dy) 0 E Z t Z h h h h h = Πx q(ξs )eq(s)ds f(s, ξs , y)π (ξs , dy) . (7.9) 0 E Combining (7.8) and (7.9) we arrive at the claim (7.7). Note that applying the shift operator

θeτn to f(τ1, ξeτ1−, ξeτ1 )1{τ1≤t} gives f(τn+1, ξeτn+1−, ξeτn+1 )1{τn+1≤t}. Using the strong Markov property of ξe and Fubini’s theorem, we can prove by induction that for all n ≥ 2,

Z t∧τn Z h i h Πe x f(τn, ξeτn−, ξeτn )1{τn≤t} = Πe x q(ξes)ds f(s, ξes, y)π (ξes, dy) . t∧τn−1 E

Thus by the above equality, Fubini’s theorem and the fact that Πe x(limn→+∞ τn = +∞) = 1, we have

" # " +∞ # X X Πe x f(τi, ξeτi−, ξeτi ) = Πe x f(τi, ξeτi−, ξeτi )1{τi≤t}

τi≤t i=1 Z t∧τn Z h = Πe x lim q(ξes)ds f(s, ξes, y)π (ξes, dy) n→+∞ 0 E Z t Z h = Πe x q(ξes)ds f(s, ξes, y)π (ξes, dy) . 0 E Hence we have proved (4.6). We next show (4.7). It is easy to see that

 !   X X Πe x  f(τi, ξeτi−, ξeτi )  g(τj, ξeτj −, ξeτj ) τi≤t τj ≤t " # X = Πe x fg(τi, ξeτi−, ξeτi )

τi≤t +∞ +∞ X X n h i + Πe x f(τi, ξeτi−, ξeτi )g(τj, ξeτj −, ξeτj )1{τj ≤t} i=1 j=i+1

53 h io + Πe x g(τi, ξeτi−, ξeτi )f(τj, ξeτj −, ξeτj )1{τj ≤t} . (7.10)

By the strong Markov property and (7.8), we have for j ≥ 2, h i Πe x f(τ1, ξeτ1−, ξeτ1 )g(τj, ξeτj −, ξeτj )1{τj ≤t} = Π f(τ , ξ , ξ )1 Π g(τ + s, ξ , ξ )1 e x 1 eτ1− eτ1 {τ1≤t} e ξeτ j−1 eτj−1− eτj−1 {τj−1+s≤t} 1 s=τ1 Z t Z h h h h h = Πx ds π (ξs , dy)q(ξs )eq(s)f(s, ξs , y)Πe y g(τj−1 + s, ξeτj−1−, ξeτj−1 )1{τj−1≤t−s} , 0 E (7.11) and for j > i ≥ 2, h i Πe x f(τi, ξeτi−, ξeτi )g(τj, ξeτj −, ξeτj )1{τj ≤t} = Π 1 Π f(τ + s, ξ , ξ )g(τ + s, ξ , ξ )1 e x {τ1≤t} e ξeτ i−1 eτi−1− eτi−1 j−1 eτj−1− eτj−1 {τj−1+s≤t} 1 s=τ1 Z t Z h h h h = Πx ds π (ξs , dy)q(ξs )eq(s) 0 E · Πe y f(τi−1 + s, ξeτi−1−, ξeτi−1 )g(τj−1 + s, ξeτj−1−, ξeτj−1 )1{τj−1≤t−s} . (7.12)

By (7.11), Fubini’s theorem, the strong Markov property of ξe, (4.6) and (7.8),

+∞ X h i Πe x f(τ1, ξeτ1−, ξeτ1 )g(τj, ξeτj −, ξeτj )1{τj ≤t} j=2 Z t Z h h h h h = Πx ds π (ξs , dy)q(ξs )eq(s)f(s, ξs , y) 0 E +∞ ! X · Πe y g(τj−1 + s, ξeτj−1−, ξeτj−1 )1{τj−1≤t−s} , j=2 "Z t Z !# h h h h h X = Πx ds π (ξs , dy)q(ξs )eq(s)f(s, ξs , y)Πe y g(τk + s, ξeτk−, ξeτk ) 0 E τk≤t−s Z t Z h h h h h = Πx ds π (ξs , dy)q(ξs )eq(s)f(s, ξs , y) 0 E Z t−s Z h · Πe y dr π (ξer, dz)q(ξer)g(r + s, ξer, z) 0 E " Z t−s Z # h = Πe x f(τ1, ξeτ1−, ξeτ1 ) Πe dr π (ξer, dz)q(ξer)g(r + s, ξer, z) 1{τ1≤t} ξeτ1 0 E s=τ1 Z t Z h = Πe x f(τ1, ξeτ1−, ξeτ1 ) dr π (ξer, dz)q(ξer)g(r, ξer, z)1{τ1≤t} . τ1 E

54 Similarly, by (7.12), Fubini’s theorem, the strong Markov property of ξe, (4.6) and (7.8), we can prove by induction that for i ≥ 1,

+∞ X h i Πe x f(τi, ξeτi−, ξeτi )g(τj, ξeτj −, ξeτj )1{τj ≤t} j=i+1 Z t Z h = Πe x f(τi, ξeτi−, ξeτi ) dr π (ξer, dz)q(ξer)g(r, ξer, z)1{τi≤t} . τi E

By this, Fubini’s theorem, the strong Markov property of ξe and (4.6), we get

+∞ +∞ X X h i Πe x f(τi, ξeτi−, ξeτi )g(τj, ξeτj −, ξeτj )1{τj ≤t} i=1 j=i+1 " +∞ Z t Z # X h = Πe x f(τi, ξeτi−, ξeτi ) dr π (ξer, dz)q(ξer)g(r, ξer, z)1{τi≤t} i=1 τi E " Z t Z # X h = Πe x f(τi, ξeτi−, ξeτi ) dr π (ξer, dz)q(ξer)g(r, ξer, z) τi E τi≤t " t−s # X Z Z = Π f(τ , ξ , ξ ) Π q(ξ )dr g(s + r, ξ , z)πh(ξ , dz) e x i eτi− eτi e ξeτ er er er i 0 E τi≤t s=τi Z t Z Z t−s Z h h = Πe x ds π (ξes, dy)q(ξes)f(s, ξes, y)Πe y dr π (ξer, dz)q(ξer)g(s + r, ξer, z) . 0 E 0 E (7.13)

Combining (7.10) and (7.13), we arrive at (4.7).

Acknowledgements

The research of Y.-X. Ren is supported by NSFC (Grant Nos. 11671017 and 11731009), and LMEQF. The research of R. Song is supported in part by a grant from the Simons Foundation (#429343). The research of T. Yang is supported by NSFC (Grant Nos. 11501029 and 11731009). The authors thank Professor Zhen-Qing Chen for his helpful comments on the construction of the concatenation process. The authors also thank the referee for very helpful comments.

References

[1] S. Albeverio and Z.-M. Ma: Perturbation of Dirichlet forms-lower semiboundedness, closability, and form cores. J. Funct. Anal., 99 (1991), 332–356.

55 [2] V. Bansaye, J.-F. Delmas, L. Marsalle and V. Tran: Limit theorems for Markov processes indexed by continous time Galton-Watson trees. Ann. Appl. Probab., 21 (2011), 2263–2314.

[3] M. Barczy and G. Pap: Asymptotic behavior of critical irreducible multi-type continuous state and continuous time branching processes with immigration. Stoch. Dyn., 16(4) (2016), Article ID 1650008, 30pages.

[4] K. Bogdan, K. Burdzy and Z.-Q. Chen: Censored stable processes. Probab. Theory Related Fields, 127 (2003), 89–152.

[5] Z.-Q. Chen: Gaugeability and conditional gaugeability. Trans. Amer. Math. Soc., 354 (2002), 4639–4679.

[6] Z.-Q. Chen and M. Fukushima: Symmetric Markov Processes, Time Changes and Boundary Theory. Princeton University Press, USA, 2012.

[7] Z.-Q. Chen, P.J. Fitzsimmons, M. Takeda, J. Ying and T.-S. Zhang: Absolute continuity of symmetric Markov processes. Ann. Probab., 32 (2004), 2067–2098.

[8] Z.-Q. Chen, P. Kim and R. Song: Heat kernel estimates for Dirichlet fractional Laplacian. J. Euro. Math. Soc., 12 (2010), 1307–1329.

[9] Z.-Q. Chen, P. Kim and R. Song: Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Related Fields, 146 (2010), 361–399.

[10] Z.-Q. Chen, P. Kim and R. Song: Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation. Trans. Amer. Math. Soc., 367 (2015), 5237–5270.

[11] Z.-Q. Chen and T. Kumagai: Heat kernel estimates for stable-like processes on d-sets. Stoch. Proc. Appl., 108 (2003), 27–62.

[12] Z.-Q. Chen, Y.-X. Ren and R. Song: L log L criterion for a class of multitype superdiﬀusions with non-local branching mechanism. Sci. China Math., 62 (2019), 1439–1462.

[13] Z.-Q. Chen, Y.-X. Ren and T. Yang: Skeleton decomposition and law of large numbers for supercritical superprocesses. Acta Appl. Math., 159(1) (2019), 225-285.

[14] Z.-Q. Chen and R. Song: Conditional gauge theorems for non-local Feynman-Kac transforms. Probab. Theory Related Fields, 125 (2003), 45–72.

[15] Z.-Q. Chen, and R. Song: Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal., 201 (2003), 262–281.

[16] G. Da Prato and J. Zabczyk: Ergodicity for Inﬁnite Dimensional Systems. Cambridge Uni- versity Press, 2003.

56 [17] D.A. Dawson, L.G. Gorostiza and Z.-H. Li: Non-local branching superprocesses and some related models. Acta Appl. Math., 74 (1993), 93–112.

[18] R. Durrett: Probability: Theory and Examples, 4th Ed. Cambridge University Press, 2010.

[19] J. Engl¨anderand A.E. Kyprianou: Local extinction versus local exponential growth for spatial branching processes. Ann. Probab., 32 (2004), 78–99.

[20] J. Engl¨anderand R.G. Pinsky: On the construction and support properties of measure valued d diﬀusions on D ⊂ R with spatially dependent branching. Ann. Probab., 27 (1999), 684–730.

[21] J. Engländer,R. Song and Y.-X. Ren: Weak extinction versus global exponential growth of total mass for superdiffusions. Ann. Inst. Henri Poincaré Probab. Stat., 52 (2016), 448–482.

[22] S.N. Evans: Two representations of a conditioned superprocess. Proc. R. Soc. Edinburgh A, 123 (1993), 959–971.

[23] S.N. Evans and N. O’Connell: Weighted occupation time for branching particle systems and a representation for the supercritical superprocesses. Canad. Math. Bull., 37 (1994), 187–196.

[24] M. Fukushima, Y. Oshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994.

[25] R. Hardy and S.C. Harris: A spine approach to branching diﬀusions with applications to Lp-convergence of martingales. In Lecture Notes in Mathematics, 1979, Springer, Berlin, pp. 281–330, 2009.

[26] S.C. Harris and M. Roberts: Measure changes with extinction. Statist. Probab. Lett., 79 (2009), 1129–1133.

[27] H. Hering: Uniform primitivity of semigroups generated perturbed elliptic diﬀerential operators. Math. Proc. Camb. Phil. Soc., 83 (1978), 261–268.

[28] N. Ikeda, M. Nagasawa and S. Watanabe: A construction of Markov processes by piecing out. Proc. Japan Academy, 42 (1966), 370–375.

[29] K. Kaleta and J. Lorinczi: Pointwise eigenfunction estimates and intrinsic ultracontractivity- type properties of Feynman-Kac semigroups for a class of L´evyprocesses. Ann. Probab., 43 (2015), 1350–1398.

[30] H. Kesten and B.P. Stigum: Alimit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat., 37 (1966), 1211–1223.

[31] P. Kim and R. Song: Intrinsic ultracontractivity of non-symmetric diﬀusion semigroups in bounded domains. Tohoku Math. J., 60 (2008), 527–547.

57 [32] H. Kunita: Sub-Markov semi-groups in Banach lattices. In Proceedings of the International Conference on Functional Analysis and Related Topics, University of Tokyo Press, 1969, 332– 343.

[33] A.E. Kyprianou, R.-L. Liu, A. Murillo-Salas and Y.-X. Ren: Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Ann. Inst. Henri Poincar´e Probab. Stat., 48 (2012), 661-687.

[34] A.E. Kyprianou and A. Murillo-Salas: Super-Brownian motion: Lp convergence of martingales through the pathwise spine decomposition. In Adavances in Superprocesses and Nonlinear PDEs, volume 38 of Springer Proceedings in Mathematics and Statistics, 2013.

[35] A.E. Kyprianou and S. Palau: Extinction properties of multi-type continuous-state branching processes. Stochastic Process. Appl., 128 (2018), 3466–3489.

[36] Z.-H. Li: Measure-valued Branching Markov Processes. Springer, Heidelberg, 2011.

[37] R.-L. Liu, Y.-X. Ren and R. Song: L log L criteria for a class of superdiﬀusons. J. Appl. Probab., 46 (2009), 479–496.

[38] S. Palau and T. Yang: Law of large numbers for supercritical superprocesses with non-local branching. Stochastic Process. Appl., 130 (2020), 1074–1102.

[39] H.H. Schaeﬀer: Banach Lattices and Positive Operators. Springer-Verlag, New York, 1974.

[40] M.J. Sharpe: General Theory of Markov Processes. Academic Press, New York, 1988.

[41] P. Stollman and J. Voigt: Perturbation of Dirichlet forms by measures. Potential Anal., 5 (1996), 109–138.