Stable Process with Singular Drift

Available online at www.sciencedirect.com ScienceDirect

Stochastic Processes and their Applications 124 (2014) 2479–2516 www.elsevier.com/locate/spa

Stable process with singular drift

Panki Kima,∗, Renming Songb a Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Building 27, 1 Gwanak-ro, Gwanak-gu, Seoul 151-747, Republic of Korea b Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Received 23 March 2013; received in revised form 11 March 2014; accepted 11 March 2014 Available online 18 March 2014

Abstract

Suppose that d ≥ 2 and α ∈ (1, 2). Let µ = (µ1, . . . , µd ) be such that each µi is a signed measure on d R belonging to the Kato class Kd,α−1. In this paper, we consider the stochastic differential equation

d Xt = dSt + d At ,

d j where St is a symmetric α-stable process on R and for each j = 1,..., d, the jth component At of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is µ j . The unique solution for the above stochastic differential equation is called an α-stable process with drift µ. We prove the existence and uniqueness, in the weak sense, of such an α-stable process with drift µ and establish sharp two-sided heat kernel estimates for such a process. ⃝c 2014 Elsevier B.V. All rights reserved.

MSC: primary 60J35; 47G20; 60J75; secondary 47D07

Keywords: Symmetric α-stable process; Gradient operator; Heat kernel; Transition density; Green function; Exit time; Levy´ system; Boundary Harnack inequality; Kato class

∗ Corresponding author. E-mail addresses: [email protected] (P. Kim), [email protected] (R. Song). http://dx.doi.org/10.1016/j.spa.2014.03.006 0304-4149/⃝c 2014 Elsevier B.V. All rights reserved. 2480 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

1. Introduction In the last decade, stochastic differential equations (SDEs) driven by jump processes have attracted the attention of many researchers as these SDEs arise naturally in various models. See, for instance, [1,3,23] and the references therein. In this paper we consider an SDE, with a singular drift which may not be absolutely con- d tinuous with respect to the Lebesgue measure on R , driven by a multidimensional symmetric α-stable process. A solution of such an SDE is called a stable process with singular drift. An example of such a process is a stable process which drifts upwards when it hits fractal-like sets (see [4, p. 792] for Brownian motion which drifts upwards when it hits fractal-like sets). The main purpose of this paper is to prove the existence and uniqueness of weak solutions to the above SDE for α ∈ (1, 2). When α = 2 (i.e., in the case of Brownian motion), this problem was solved by Bass and Chen in [4], and later in [18,19] we established the boundary Harnack principle and sharp two-sided estimates on the densities of such processes. When α ∈ (1, 2) and the drifts are absolutely continuous with respect to the Lebesgue measure, sharp two-sided estimates on the heat kernels corresponding to the generators of the solutions to such SDEs were d , obtained in R [8,16,17] and in bounded C1 1 open sets [9]. In this paper, we first establish sharp two-sided heat kernel estimates and then use these estimates to prove the existence and uniqueness of α-stable processes with singular drifts. In [4], the existence and uniqueness of Brownian motions with singular drifts were established without using sharp estimates on the densities of these processes. Throughout this paper we assume d ≥ 2, α ∈ (1, 2). For functions f and g, the notation “ f ≍ g” means that there exist constants c2 ≥ c1 > 0 such that c1 g ≤ f ≤ c2 g. We will use dx d to denote the Lebesgue measure in R . Here and in the sequel, we use := as a way of definition. For a, b ∈ R, a ∧ b := min{a, b} and a ∨ b := max{a, b}. d Throughout this paper we assume that S is a (rotationally) symmetric α-stable process on R . The infinitesimal generator of S is ∆α/2 := −(−∆)α/2 which can be defined by  A(d, α) ∆α/2u x := u y − u x dy ( ) lim ( ( ) ( )) d+α , r→0 B(x,r)c |x − y| = α−1 −d/2 d+α − α −1 where A(d, α) α2 π Γ ( 2 )Γ (1 2 ) . We will use p(t, x, y) to denote the transition density of S (or equivalently, the heat kernel of ∆α/2). It is well-known (see, e.g., [6]) that  t  p(t, x, y) ≍ t−d/α ∧ on (0, ∞) × d × d . (1.1) |x − y|d+α R R It follows from [8, Lemma 5] that  t  |∇ p(t, x, y)| ≍ |x − y| t−(d+2)/α ∧ on (0, ∞) × d × d . (1.2) x |x − y|d+2+α R R For any λ ≥ 0, we define  ∞ Rλ(x, y) = e−λt p(t, x, y)dt. (1.3) 0

Then, there exists a constant C1 = C1(d, α) > 1 such that for λ ≥ 0, −1 | − |α−d ∧ −2| − |−d−α ≤ λ C1 ( x y (λ x y )) R (x, y) α−d −2 −d−α ≤ C1(|x − y| ∧ (λ |x − y| )) (1.4) P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2481 and −1 | − |α−d−1 ∧ −2| − |−d−α−1 ≤ |∇ λ | C1 ( x y (λ x y )) x R (x, y) α−d−1 −2 −d−α−1 ≤ C1(|x − y| ∧ (λ |x − y| )). (1.5) (For example, see [8, Lemmas 7 and 9].) Note that, in particular, when λ = 0 we have 0 α−d−1 |∇x R (x, y)| ≍ |x − y| . Thus,  | f |(y)  dy ≍ | f (y)||∇ R0(x, y)|dy. sup d+1−α sup x d |x − y| d x∈R B(x,r) x∈R B(x,r)

d Definition 1.1. For any function f on R , we define for r > 0,  | f |(y) Mα(r) = dy. f sup d+1−α d |x − y| x∈R B(x,r) d α = A function f on R is said to belong to the Kato class Kd,α−1 if limr↓0 M f (r) 0. By a signed measure ν we mean in this paper the difference of two nonnegative σ-finite d measures ν1 and ν2 in R . Note that there is an increasing sequence of subsets {Fk, k ≥ 1} d whose union is R so that ν1(Fk) + ν2(Fk) < ∞ for every k ≥ 1. So when restricted to each Fk, ν is a finite signed measure. Consequently, the positive and negative parts of ν are well d + − defined on each Fk and hence on R , which will be denoted as ν and ν , respectively. We use |ν| = ν+ + ν− to denote the total variation measure of ν. In this paper we take such an extended view of signed measures and extend Definition 1.1 to signed measures.

d Definition 1.2. For any signed measure ν on R , we define for r > 0,  |ν|(dy) Mα(r) = . ν sup d+1−α d |x − y| x∈R B(x,r) d α A signed measure ν on R is said to belong to the Kato class Kd,α−1 if limr↓0 Mν (r) = 0. We d d say that a d-dimensional vector valued signed measure µ = (µ1, . . . , µ ) on R belongs to the j Kato class Kd,α−1 if each µ belongs to the Kato class Kd,α−1. Since 1 < α < 2, using Holder’s¨ inequality, it is easy to see that for every p > d/(α − 1), ∞ d p d L (R ; dx) + L (R ; dx) ⊂ Kd,α−1. Moreover, as we will see from (2.1) below, every signed measure in the Kato class Kd,α−1 is Radon. Throughout this paper we will assume that µ = (µ1, . . . , µd ) is a vector valued signed Radon d measure on R belonging to the Kato class Kd,α−1. d Let ϕ(x) be a non-negative smooth radial function in R such that supp[ϕ] ⊂ B(0, 1) and  nd n −n  ϕ(x)dx = 1. Let ϕn(x) := 2 ϕ(2 x). We note that supp[ϕn] ⊂ B(0, 2 ) and ϕn(x)dx = j d 1. We will fix ϕ and ϕn throughout this paper. For signed Radon measures µ on R with 1 ≤ j ≤ d, define  j j Un (x) = ϕn(x − y)µ (dy) (1.6) d R 2482 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 and set j j µn(dx) = Un (x)dx. (1.7) 1 d 1 d We write Un(x) for (Un (x), . . . , Un (x)) and µn for (µn, . . . , µn ). We also fix Un(x) and µn j throughout this paper. It follows from (2.1) below that when µ ∈ Kd,α−1, j = 1,..., d, each j d Un is a bounded and smooth function on R . d d Let Ω = D([0, ∞), R ) be the family of cadl` ag` functions from [0, ∞) to R , equipped with the Skorohod topology. We will use Xt to denote the coordinate map on Ω : Xt (ω) = ω(t) and Ft to denote the σ-field σ{Xs : s ≤ t}. F∞ stands for the σ-field σ{Xs : s ≥ 0}. It is well-known (see, for instance, [11, Proposition 3.7.1]) that F∞ coincides with the Borel σ-field on Ω = d d D([0, ∞), R ). Here is the definition of an α-stable process with drift on R , which is motivated by the definition of a Brownian motion with singular drift introduced in[4].

Definition 1.3. Given a d-dimensional vector valued signed measure µ = (µ1, . . . , µd ) belong- d d ing to Kd,α−1 and x ∈ R , an α-stable process with drift µ on R is a probability measure P on (Ω, F∞) such that

Xt = x + St + At (1.8) where =  t (a) At limn→∞ 0 Un(Xs)ds uniformly in t over finite intervals, where the convergence is in probability and Un is defined in (1.6); { }  t | | ∞ (b) there exists a subsequence nk such that supk 0 Unk (Xs) ds < a.s. for each t > 0; d (c) under P, S0 = 0 and St is a symmetric α-stable process in R . d An α-stable process with drift µ on R is also called a weak solution to the SDE (1.8). Here is the main result of this paper.

Theorem 1.4. (a) There exists one and only one weak solution to (1.8). This unique solution is conservative. d (b) Let (X, Px ) be the solution in (a) with X0 = x. Then (X, Px , x ∈ R ) forms a strong µ d d Markov process with a jointly continuous transition density p (t, x, y) on (0, ∞)×R ×R . Furthermore, each component A j of A is a continuous additive functional of X of finite variation with respect to X whose Revuz measure is µ j , in the sense that, for every t > 0 and d bounded Borel measurable function f on R ,  t  t  j µ j d Ex f (Xs)d As = p (s, x, y) f (y)µ (dy)ds, x ∈ R . d 0 0 R d (c) (X, Px , x ∈ R ) is a Feller process with the strong Feller property. Furthermore, for every T > 0 there is a positive constant c > 1 depending only on α, d and T and the rate at which d Mα (r) goes to zero such that for all (t, x, y) ∈ (0, T ] × d × d , j=1 µ j R R  t   t  c−1 t−d/α ∧ ≤ pµ(t, x, y) ≤ c t−d/α ∧ . |x − y|d+α |x − y|d+α

The organization of this paper is as follows. Section2 contains some basic properties of µ measures in the Kato class Kd,α−1. In Section3 we construct the heat kernel p (t, x, y) of ∆α/2 + µ · ∇ and establish sharp two-sided estimates on pµ(t, x, y). In Section4, we study P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2483 the convergence of the perturbed semigroups and resolvents. In Section5 we use results of the previous sections to establish the existence and uniqueness of stable processes with singular drifts. Some details of the proof of the uniqueness are spelled in the Appendix. After we have finished the first version of this paper, we were informed that veryrecently in [10] Chen and Wang studied on a related topic. In that paper, they showed that, when µ(dx) = b(x)dx is absolutely continuous and belongs to the Kato class Kd,α−1, the martingale α/2 problem for L = ∆ + b · ∇ is well posed and the SDE d Xt = dSt + b(Xt )dt has a unique weak solution in the “classical” sense. As remarked in [10], our Theorem 1.4 does not cover these results in [10]. In the remainder of this paper, the constants C1, C2, C3 will be fixed. The lower case constants c1, c2,... can change from one appearance to another. The dependence of the constants on the dimension d ≥ 2 and the stability index α ∈ (1, 2) will not be always mentioned explicitly. For a d Borel set A ⊂ R , we also use |A| to denote its Lebesgue measure. The space of continuous d d d ∞ d functions on R will be denoted as C(R ), while C∞(R ) and Cc (R ) denote the space d of continuous functions on R that vanish at infinity and the space of smooth functions with compact supports respectively. 2. Preliminary

In this section we collect some basic properties of signed measures in the Kato class Kd,α−1. Recall that we always assume that d ≥ 2, α ∈ (1, 2). α If ν ∈ Kd,α−1, then for any r > 0, Mν (r) < ∞ and  r d+1−α|ν|(dy) |ν|(B(x, r)) ≤ = r d+1−α Mα(r). sup sup d+1−α ν (2.1) d d | − |≤ |x − y| x∈R x∈R x y r In particular, (2.1) implies that for every ν ∈ Kd,α−1 and m ≥ 1, the function x → Vm(x) :=  d ϕm(x − y)ν(dy) is bounded and smooth. R The following result is basically [22, Lemma 1.1].

d Lemma 2.1. Let ν be a Radon measure on R and β a positive constant. There exists c = c(d, β) > 0 such that  ν(dy) ≤ c r −d−β B x r ∀r sup d+β sup ν( ( , )), > 0. d c |x − y| d x∈R B(x,r) x∈R d d Proof. Suppose that sup d ν(B(x, r)) < ∞. For any x ∈ , we can cover with x∈R √ R R non-overlapping cubes of side length (2/ d)r with one of the cubes centered at x. We divide the cubes into layers, counting from the cube at x. For any y in a cube in the n-th layer, 2n − 1 2 2n − 1 |x − y| ≥ √ r = √ r. 2 d d There are at most (2n + 1)d − (2n − 1)d cubes in the n-th layer. Each cube can be covered by a ball of radius r. Thus ∞ −(d+β)  ν(dy)    (2n − 1)r  ≤ n + d − n − d √ B z r d+β (2 1) (2 1) sup ν( ( , )) c |x − y| d B(x,r) n=0 d z∈R −(d+β) ∞ −  r   (2n + 1)d 1 ≤ d √ B z r 2 d+β sup ν( ( , )). (2n − 1) d d n=0 z∈R 2484 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

Combining Lemma 2.1 and (2.1) we immediately get the following corollary.

Corollary 2.2. There exists c = c(d, β) > 0 such that for every ν ∈ Kd,α−1,  |ν|(dy) ≤ cr 1−α−β Mα r ∀r sup d+β ν ( ), > 0. d c |x − y| x∈R B(x,r) The first part of the following result is a sharpening of[9, Lemma 2.2].

Lemma 2.3. If ν ∈ Kd,α−1, then there exists c = c(d, α) > 0 such that for all t > 0,  t  1/α α 1/α sup p(s, x, y)|ν|(dy)ds ≤ ct Mν (t ). d d x∈R 0 R

1/α 1/α d+1−α α 1/α sup |ν|(B(x, s )) ≤ (s ) Mν (t ). (2.3) d x∈R On the other hand, applying Corollary 2.2 we have that for any s ≤ t,  |ν|(dy) dy ≤ c s1/α−2 Mα t1/α sup d+α 2 ν ( ). (2.4) d 1/α c |y − x| x∈R B(x,s ) Combining (2.2)–(2.4), we conclude that  t   t 1/α−1 α 1/α 1/α α 1/α sup p(s, x, y)|ν|(dy)ds ≤ c3 s dsMν (t ) = c3αt Mν (t ). d d x∈R 0 R 0 When ν is absolutely continuous with respect to the Lebesgue measure, the following result is contained in the proof of [8, Lemma 14].

Lemma 2.4. If ν ∈ Kd,α−1, then  t  lim sup |∇x p(s, x, y)| |ν|(dy)ds = 0. t→0 d d x∈R 0 R

Proof. This proof is the same as that of [8, Lemma 14], so we omit the details.

Lemma 2.5. Suppose that ν belongs to the Kato class Kd,α−1. For n ≥ 1, definen V (x) :=  ϕn(x − y)ν(dy). Then each Vn belongs to the Kato class Kd,α−1 and Mα (r) ≤ Mα(r) r > 0. (2.5) Vn ν

Proof. See the proof of [4, Proposition 3.6]. P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2485

3. Sharp heat kernel estimates

In this section, we will assume α ∈ (1, 2) and construct the fundamental solution pµ(t, x, y) of L := ∆α/2 + µ · ∇. Then we establish sharp two-sided estimates of pµ(t, x, y), which will prove Theorem 1.4(c). i Recall that Un is defined in (1.6). Since for each n, the components of Un are bounded smooth functions, by using Picard’s iteration, one can easily show that, for any α ∈ (0, 2), the stochastic differential equation n n d Xt = dSt + Un(Xt )dt has a unique strong solution. This unique solution is a strong Markov process and its infinitesimal α/2 n generator is Ln := ∆ + Un · ∇. The unique solution X is called an α-stable process with n n drift Un. We will use Px to denote the probability on (Ω, F∞) induced by the law of X when n = X0 x. In the remainder of this paper, we will always assume α ∈ (1, 2). The fundamental solution U p n (t, x, y) of Ln and p(t, x, y) are related by the following Duhamel’s formula (see [8]):  t  Un Un p (t, x, y) = p(t, x, y) + p (s, x, z) Un(z) · ∇z p(t − s, z, y)dzds. (3.1) d 0 R Applying the above formula repeatedly, one expects that pUn (t, x, y) can be expressed as an infinite series in terms of p and its derivatives. This motivates the following definition. Define Un := ≥ p0 (t, x, y) p(t, x, y) and for k 1,  t  Un := Un · ∇ − pk (t, x, y) pk− (s, x, z) Un(z) z p(t s, z, y)dz. (3.2) d 1 0 R In the rest of this paper, the meaning of the phrase “depending on µ only via the rate at which α Mµ(r) goes to zero” is that the statement is true for any d-dimensional vector valued signed d measure ν on R with d d  α ≤  α Mν j (r) Mµ j (r) for all r > 0. j=1 j=1 Using (2.5),[8, Theorem 1, Lemmas 15 and 23] and [9, Proposition 2.4], we have the follow- d d ing result. Recall that we denote by C∞(R ) the space of continuous functions on R that vanish ∞ d at infinity and Cc (R ) the space of smooth functions with compact supports.

Theorem 3.1. (i) There exist T0 > 0 and c1 > 1 depending on µ only via the rate at which d Mα (r) goes to zero such that for every n ≥ 1, ∞ pUn (t, x, y) converges locally i=1 µi k=0 k d d U uniformly on (0, T0] × R × R to a positive jointly continuous function p n (t, x, y) and d d that on (0, T0] × R × R ,  t   t  c−1 t−d/α ∧ ≤ pUn (t, x, y) ≤ c t−d/α ∧ . (3.3) 1 |x − y|d+α 1 |x − y|d+α

 Un d Moreover, d p (t, x, y)dy = 1 for every n ≥ 1, t ∈ (0, T0] and x ∈ . R R (ii) For every n ≥ 1, the function pUn (t, x, y) defined in (i) can be extended uniquely to a d d positive jointly continuous function on (0, ∞) × R × R so that for all s, t ∈ (0, ∞) and 2486 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

d d  Un (x, y) ∈ × , d p (t, x, y)dy = 1 and R R R  pUn (s + t, x, y) = pUn (s, x, z)pUn (t, z, y)dz. (3.4) d R (iii) If we define  Un Un Pt f (x) := p (t, x, y) f (y)dy, (3.5) d R ∞ d d then for any f ∈ Cc (R ) and g ∈ C∞(R ),   −1 Un lim t (Pt f (x) − f (x))g(x)dx = Ln f (x) g(x)dx. t↓0 d d R R U Thus p n (t, x, y) is the fundamental solution of Ln in the distributional sense. ∞ Un We remark here that, in fact, by [17, Theorem 2] the series k=0 pk (t, x, y) in Theo- d d rem 3.1(i) converges on (0, ∞) × R × R . Un By [9, Proposition 2.3], for every n ≥ 1, the operators {Pt ; t ≥ 0} defined by (3.5) form n d a Feller semigroup and thus define a conservative Feller process Y in R . It follows from n [9, Theorem 2.5] that Y is a solution to the martingale problem associated with Ln. Being n a strong solution, X is also a solution to the martingale associated with Ln. Since the martingale problem associated with Ln is unique because Un are bounded and α ∈ (1, 2) (see [20, Theorem 2]), Y n and X n have the same law. Thus pUn (t, x, y) is the transition density of X n. For any λ > 0, we will use Gµn,λ to denote the λ-potential kernel of X n:  ∞ Gµn,λ(x, y) = GUn,λ(x, y) := e−λt pUn (t, x, y)dt. 0

By [8, Lemma 13] and its proof, there exists a constant C2 > 0 such that   t − | | |∇ | ≤ p(t s, x, z) Un(z) z p(s, z, y) dsdz C2 p(t, x, y)NUn (t), (3.6) d R 0 where d   t   :=  | j | | − |−d−1 ∧ −(d+1)/α NUn (t) sup Un (z) w z s dsdz. d d j=1 w∈R R 0

We remark that the constant C2 depends only on d and α (independent of n, t and µ) and is not the same constant C4 from [8, Lemma 13]. Let d   t   −d−1 −(d+1)/α j Nµ(t) := sup |w − z| ∧ s ds|µ |(dz). d d j=1 w∈R R 0 By following the proof of [8, Lemma 13] line by line, we also have

d   t  i p(t − s, x, z)|∇z p(s, z, y)|ds|µ |(dz) ≤ C2 p(t, x, y)Nµ(t). (3.7) d j=1 R 0

Lemma 3.2. For all t > 0, we have ≤ NUn (t) Nµ(t). (3.8) P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2487

Proof. By a change of variable, we have for j = 1,..., d,   t j  −d−1 −(d+1)/α |Un (z)| |w − z| ∧ s dsdz d R 0   t    j   −d−1 −(d+1)/α =  ϕn(z − y)µ (dy) |w − z| ∧ s dsdz d d R 0  R    t  j  −d−1 −(d+1)/α ≤ ϕn(z − y)|µ |(dy) |w − z| ∧ s dsdz d d R 0 R    t  −d−1 −(d+1)/α j = ϕn(v) |w − v − y| ∧ s dsdv|µ |(dy) d d R R 0    t   −d−1 −(d+1)/α j = ϕn(v) |w − v − y| ∧ s ds|µ |(dy) dv. d d R R 0 Thus,   t j  −d−1 −(d+1)/α sup |Un (z)| |w − z| ∧ s dsdz d d w∈R R 0      t   −d−1 −(d+1)/α j ≤ ϕn(v)dv sup |w − y| ∧ s ds|µ |(dy) d d d R w∈R R 0   t  −d−1 −(d+1)/α j = sup |w − y| ∧ s ds|µ |(dy). d d w∈R R 0

Lemma 3.2 and (3.7) imply that

t   p(t − s, x, z)|∇z p(s, z, y)| sup |Un(z)|dsdz ≤ C2Nµ(t), (3.9) d n R 0 p(t, x, y) which is finite and goes to zero as t → 0 by [8, Corollary 12] and its proof. The proof of the next lemma is similar to that of [18, Lemma 3.2].

Lemma 3.3. Suppose ν belongs to the Kato class Kd,α−1. Suppose that A is a compact subset of d R such that Hd−1(A) < ∞, where Hd−1 is the (d − 1)-dimensional Hausdorff measure. Then we have |ν|(A) = 0. Proof. For each r > 0, let N(A, r) be the smallest number of balls with radius r needed to cover A, i.e.,  k   d N(A, r) := min k : A ⊂ B(xn, r) for some x1,..., xk ∈ R . n=1

So for each r > 0, there exists a sequence {xn}1≤n≤N(A,r) such that N(A,r)  |ν|(A) ≤ |ν|(B(xn, r)). n=1 Using (2.1), we get

d+1−α α |ν|(A) ≤ r N(A, r) Mν (r). (3.10) 2488 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

Let  d  A(r) := x ∈ R : dist(x, A) ≤ r . It is well-known (see, for instance, [21, (5.4) and (5.6)]) that there exists a positive number c1 = c1(d) such that d r N(A, r) ≤ c1 |A(r)|. (3.11)

Since Hd−1(A) < ∞, it follows from [12, Theorem 3.2.39] that there exists c2 = c2(d) > 0 such that 1 lim |A(r)| = c2Hd−1(A) < ∞. (3.12) r↓0 r Thus, combining (3.10)–(3.12), we have

1 α α |ν|(A) ≤ c1 lim |A(r)|Mν (r) = c1c2Hd−1(A) lim Mν (r) = 0. r↓0 r α−1 r↓0

Using our Lemma 3.3, instead of [18, Lemma 3.2], the proof of the next lemma is the same as that of [18, Lemma 3.3]. So we omit the proof.

d Lemma 3.4. Let 0 < T0 < T1 < ∞. Suppose K1 is a compact subset of R and D1 is a bounded domain with smooth boundary ∂ D1. Then for any signed measure ν ∈ Kd,α−1, continuous function f on [T0, T1] × K1 × K1 × D1, we have     lim sup  f (t, x, y, z)(Vn(z)dz − ν(dz)) = 0, n↑∞ (t,x,y)∈[T1,T2]×K1×K1  D1   where Vn(x) := ϕn(x − y)ν(dy). Recall that Rλ(x, y) is defined in (1.3) and satisfies the estimates in (1.4) and (1.5).

Lemma 3.5. Suppose that ν ∈ Kd,α−1 has compact support, then for every λ ≥ 0,    λ λ  lim sup  ∇x R (x, y)Vn(y)dy − ∇x R (x, y)ν(dy) = 0, n→∞ d  d d  x∈R R R  where Vn(x) := ϕn(x − y)ν(dy). Furthermore, the rate of convergence depends on µ only via the rate at which Mα (r) tends to zero. µ j Proof. Without loss of generality, we assume that the support of ν is contained in B(0, R) for some R ≥ 1. It follows from (1.5) and Lemma 2.5 that, for any ε > 0, we can choose m > 3 large and r ∈ (0, 1/4) small so that  λ α−d−1 sup |∇x R (x, y)| |ν|(dy) ≤ c1((m − 1)R) |ν|(B(0, R)) ≤ ε/8, c d x∈B(0,m R) R  λ α−d−1 sup sup |∇x R (x, y)| |Vn(y)|dy ≤ c1((m − 2)R) |ν|(B(0, R)) ≤ ε/8, c d n≥1 x∈B(0,m R) R  λ α sup |∇x R (x, y)| |ν|(dy) ≤ c1 Mν (r) ≤ ε/8, d x∈R B(x,r) P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2489 and  λ α sup sup |∇x R (x, y)| |Vn(y)|dy ≤ c1 Mν (r) ≤ ε/8. ≥ d n 1 x∈R B(x,r) d Fix such an ε > 0, m and r > 0. Define Dx := B(0, 2R) \ B(x, r) and An,x := {w ∈ R : −n −n dist(w, ∂ Dx ) ≤ 2 }, n ≥ 1, and let N(An,x ) be the smallest number of balls of radius 2 needed to cover An,x , i.e.,  k   −n d N(An,x ) := min k : An,x ⊂ B(xl , 2 ) for some x1,..., xk ∈ R . l=1 ≥ ∈ d { x,n} Thus for each n 1 and x R , there exists a sequence yl 1≤l≤N(An,x ) such that

N(An,x ) | | ≤  | | x,n −n ν (An,x ) ν (B(yl , 2 )). l=1 Using (2.1), we get

−n d+1−α α −n |ν|(An,x ) ≤ (2 ) N(An,x ) Mν (2 ). (3.13)

As in (3.11) (and using the triangle inequality), there exists a positive number c2 = c2(d) such that −n d d −n (2 ) N(An,x ) ≤ c2 |{z ∈ R : dist(z, An,x ) ≤ 2 }| ≤ c2 |An−1,x |. (3.14) Clearly, n sup sup 2 |An−1,x | = c3 < ∞. (3.15) ≥ d n 1 x∈R Thus, since ν ∈ Kd,α−1, combining (1.5) and (3.13)–(3.15), we have that there exists n0 such that for all n ≥ n0, λ α−d−1 sup sup |∇x R (x, y)|( sup |ν|(An,x )) ≤ c4r sup |ν|(An,x ) d ∈ d d x∈R y Dx x∈R x∈R α−d−1 −n(2−α) α −n ≤ c2c3c4r 2 Mν (2 ) ≤ ε/8. (3.16) λ λ We extend ∇x R (x, ·)|Dx to be zero off Dx and denote the extension by f (x, ·). Note that by (3.16),    ∗ λ · − λ   ((ϕn f (x, ))(w) f (x, w))ν(dw) An,x   λ ≤ |∇x R (x, y)|ϕn(w − y)dy|ν|(dw)ε/8 < ε/4. An,x y∈Dx

Therefore, by Fubini’s theorem, we have for all x ∈ B(0, m R) and n ≥ n0,    λ   λ   ∇x R (x, y)(Vn(y)dy − ν(dy)) =  f (x, y)(Vn(y)dy − ν(dy))    d  Dx R     λ λ  =  ϕn(w − y)ν(dw) f (x, y)dy − f (x, w)ν(dw)  d d d  R R R  λ λ  =  ((ϕn ∗ f (x, ·))(w) − f (x, w))ν(dw)  d  R 2490 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516  λ λ ≤ |(ϕn ∗ f (x, ·))(w) − f (x, w)| |ν|(dw) (Dx \An,x )∪An,x  λ λ ε ≤ |(ϕn ∗ f (x, ·))(w) − f (x, w)| |ν|(dw) + Dx \An,x 4 λ λ ε ≤ |ν|(B(0, 2R)) sup |(ϕn ∗ f (x, ·))(w) − f (x, w)| + x∈B(0,m R),w∈Dx \An,x 4   λ ≤ |ν|(B(0, 2R)) sup  ϕn(z)( f (x, w + z)  −n x∈B(0,m R),w∈Dx \An,x B(0,2 )  ε − f λ(x, w))dz +  4 ε ≤ |ν|(B(0, 2R)) sup | f λ(x, w + z) − f λ(x, w)| + . −n x∈B(0,m R),w∈Dx ,|z|<2 4 The first term in the last line above goes to 0as n → ∞ by the uniform continuity of λ (x, y) → ∇x R (x, y) in the compact set {(x, w) : x ∈ B(0, m R) × (B(0, 4R) \ B(x, r/2))}. The assertion about the rate of convergence is clear from the argument above.

Lemma 3.6. There exists a constant c = c(d, α) > 0 such that for every ν ∈ Kd,α−1, m > 1 and r > 0  −d−α 1−2α α −α sup |y − x| |ν|(dy) ≤ cr Mν (r)m . d c x∈R B(x,mr)

Proof. Note that ∞   |y − x|−d−α|ν|(dy) ≤ (kr)−(d+α)ν(A(x, kr,(k + 1)r)), c B(x,mr) k=m where A(x, kr,(k + 1)r) = B(x,(k + 1)r) \ B(x, kr). Each A(x, kr,(k + 1)r) can be covered d−1 by c1k balls of radius r, thus by (2.1)  ∞ −d−α  −(d+α) d−1 d+1−α α sup |y − x| |ν|(dy) ≤ c2 (kr) k r Mν (r) d c x∈R B(x,mr) k=m ∞ 1−2α α  −1−α = c2r Mν (r) k k=m 1−2α α −α ≤ c3r Mν (r)m .

Lemma 3.7. There exists a constant c = c(d, α) > 0 such that for every ν ∈ Kd,α−1, t > d 0, x ∈ R , m ≥ 1, n ≥ 1 and 1 ≤ j ≤ d  t  sup p(t − s, x, z)|∇z p(s, z, y)|(|Vn(z)|dz + |ν|(dz))ds |x|,|y|

Proof. First, we claim that there exists a positive constant c1 depending only on d and α such d | | | | that for any measure ν on R , t, m > 0 and x , y < m,  t  − |∇ | p(t s, x, z) z p(s, z, y) ν(dz)ds 0 |x−z|≥4m  t  ≤ −1/α c1 p(t, x, y) sup s p(s, u, z)ν(dz)ds. (3.17) |u| 4m, we have |y−z| > |x−z|−|x−y| > 4m−|x−y| > 2m. Therefore, (3.17) follows by taking the supremum in (3.18) over |x|, |y| < m. −d−α d Since p(s, x, v) ≤ c3s|x − v| , we have that for any x ∈ R ,  t  −1/α s p(s, x, v)(|Vn(v)|dv + |ν|(dv))ds 0 |v−x|≥2m  2−1/α −d−α ≤ c4t |x − v| (|Vn(v)|dv + |ν|(dv)). |v−x|≥2m d Since ϕ is a non-negative radial function supported by B(0, 1), we have for any x ∈ R and m > 2,    −d−α −d−α |x − v| |Vn(v)|dv = |x − v| ϕn(v − z)|ν|(dz)dv |v−x|≥2m |v−x|≥2m |v−z|≤1   −d−α ≤ ϕn(v − z)|x − v| dv|ν|(dz). |z−x|≥3m/2 |v−z|≤1 Using the change of variable v = z − w in the inner integral we get that   −d−α ϕn(v − z)|x − v| dv|ν|(dz) |z−x|≥3m/2 |v−z|≤1   −d−α = ϕn(w)|x − z + w| dw|ν|(dz) |z−x|≥3m/2 |w|≤1   −d−α = ϕn(w) |x − z + w| |ν|(dz)dw |w|≤1 |z−x|≥3m/2  ≤ |x − u|−d−α|ν|(du). |u−x|≥m 2492 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

Thus,  t  −1/α sup s p(s, x, v)(|Vn(v)|dv + |ν|(dv))ds |x|

Choose T1 > 0 small such that 1 C2Nµ(t) < for every t ≤ T1. (3.22) 2

Then by (3.21), for t ∈ (0, T1], we can define ∞ µ :=  µ p (t, x, y) pk (t, x, y), k=0 and for t ∈ (0, T1], pUn (t, x, y) ∨ pµ(t, x, y) ≤ 2p(t, x, y). (3.23)

In the remainder of this section, T1 stands for the constant T1 defined in (3.22). Recall that we 1 d fixed µ = (µ , . . . , µ ) belonging to Kd,α−1.

d Lemma 3.8. For any compact subsets K1 and K2 of R , and T0 ∈ (0, T1], we have for all k ≥ 0, lim sup |pUn (t, x, y) − pµ(t, x, y)| = 0. n→∞ k k (t,x,y)∈[T0,T1]×K1×K2

Proof. Without loss of generality, we may assume that T0 < 2. We will prove this lemma by induction. This lemma is clearly valid for k = 0. We assume that the lemma is true for k, which µ in particular implies that pk (t, x, y) is jointly continuous. P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2493

It suffices to prove the assertion for k + 1 when K1 and K2 are equal and are subsets of B(0, m/2) for some m > 1. We let K := K1 = K2. Since |x| < m, we have |x − z| > |z| − |x| > 4m if |z| ≥ 5m. If t ∈ [T0, T1], by (3.21) and Lemma 3.7, d  t  :=  | Un | | − | | j | A(n, t, x, y) pk (s, x, z) ∂z j p(t s, z, y) Un (z) dzds j=1 0 |z|≥5m  t   + | µ | | − | | j | pk (s, x, z) ∂z j p(t s, z, y) µ (dz)ds 0 |z|≥5m d   t  ≤ c1 p(s, x, z)|∇z p(t − s, z, y)| j=1 0 |x−z|≥4m j j × (|Un (z)|dz + |µ |(dz))ds d d ≤ 2−(d+1)/α  α −α ≤  α −α c2t Mµ j (1)m c3 Mµ j (1)m j=1 j=1 where c3 depends only on T0, T1, d and α. Thus, for any given ε > 0, we can choose m large enough so that sup A(n, t, x, y) < ε. (n,t,x,y)∈N×[T0,T1]×K ×K

We fix such an m and, for any (δ, n, t, x, y) ∈ (0, T0/2) × N × [T0, T1] × K × K , we define d   δ = :=  | Un | | − | | j | I I(n, δ, t, x, y) pk (s, x, z) ∂z j p(t s, z, y) Un (z) dsdz j=1 B(0,5m) 0   δ  + | µ | | − | | j | pk (s, x, z) ∂z j p(t s, z, y) ds µ (dz) , B(0,5m) 0 d   t = :=  | Un | | − | | j | II II(n, δ, t, x, y) pk (s, x, z) ∂z j p(t s, z, y) Un (z) dsdz j=1 B(0,5m) t−δ   t  + | µ | | − | | j | pk (s, x, z) ∂z j p(t s, z, y) ds µ (dz) , B(0,5m) t−δ d   t−δ = :=   µ − j III III(n, δ, t, x, y)  pk (s, x, z)∂z j p(t s, z, y)Un (z)dsdz j=1  B(0,5m) δ   t−δ  − µ − j  pk (s, x, z)∂z j p(t s, z, y)dsµ (dz) , B(0,5m) δ  and IV = IV(n, δ, t, x, y) d   t−δ :=  | Un − µ | | − | | j | pk (s, x, z) pk (s, x, z) ∂z j p(t s, z, y) Un (z) dsdz. j=1 B(0,5m) δ Then we have | Un − µ | ≤ + + + + pk+1(t, x, y) pk+1(t, x, y) ε I II III IV. 2494 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

Since δ < T0/2, by (1.2), (3.21) and (3.22), we have d   δ − d+2  j j I ≤ c3(T0/2) α sup p(s, w, z)(|Un (z)|dz + |µ |(dz))ds. d j=1 w∈R B(0,5m) 0

Here we used the inequalities T0/2 = T0 − T0/2 ≤ t − T0/2 < t − s. Similarly, by (1.1), (3.21) and (3.22), we have d   δ − d  j j II ≤ c4(T0/2) α sup |∇z p(s, z, w)|(|Un (z)|dz + |µ |(dz))ds. d j=1 w∈R B(0,5m) 0

By Lemmas 2.3–2.5, we choose δ < T0/2 such that I and II are less than or equal to ε/8 for every n ≥ 1. Now, we fix such δ and estimate III. Let  t−δ := µ − = f j (t, x, y, z) pk (s, x, z)∂z j p(t s, z, y)ds j 1,..., d. δ µ − By the continuity of pk (s, x, z) and ∂z j p(t s, z, y), f j (t, x, y, z) is uniformly continuous on [T0, T1] × K1 × K1 × B(0, 5m). Therefore, by Lemma 3.4, lim sup III(n, δ, t, x, y) = 0. n→∞ (t,x,y)∈[T0,T1]×K ×K Finally, we estimate IV. From (1.2), we easily see that sup IV(n, δ, t, x, y) (t,x,y)∈[T0,T1]×K ×K

d   T1 ≤ −k −(d+2)/α  | Un − µ | | j | C22 δ sup pk (s, x, z) pk (s, x, z) Un (z) dsdz x∈K j=1 B(0,5m) δ  ≤ −k −(d+2)/α | Un C22 δ T1 sup pk (s, x, z) (s,x,z)∈[δ,T1]×K ×B(0,5m)  d  − µ |  | j | pk (s, x, z) Un (z) dz. j=1 B(0,5m) Using (2.1) and Lemma 2.5, we have that −k −(d+2)/α sup IV(n, δ, t, x, y) ≤ C22 δ T1 (t,x,y)∈[T0,T1]×K ×K   d × | Un − µ | d+1−α  α sup pk (s, x, z) pk (s, x, z) (5m) Mµ j (5m). (s,x,z)∈[δ,T1]×K ×B(0,5m) j=1 Thus, by the induction hypotheses, lim sup IV(n, δ, t, x, y) = 0. n→∞ (t,x,y)∈[T0,T1]×K ×K

µ ∞ × d × d Lemma 3.8 implies that pk (t, x, y) is jointly continuous in (0, ) R R , which also follows from the proof of [8, Lemma 14]. It follows from Lemma 3.8 and the definitions of pµ and pUn that P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2495

Theorem 3.9. As n → ∞, pUn (t, x, y) converges to pµ(t, x, y), uniformly on any compact d d subset of (0, ∞) × R × R . Combining Theorems 3.1 and 3.9 with the argument on [18, pp. 655–656], we immediately get

Theorem 3.10. (i) There exist T0 > 0 and c1 > 1 depending on µ only through the rate at which d Mα (r) goes to zero such that ∞ pUn (t, x, y) converges locally uniformly j=1 µ j k=0 k d d µ on (0, T0] × R × R to a positive jointly continuous function p (t, x, y) and that on d d (0, T0] × R × R ,  t   t  c−1 t−d/α ∧ ≤ pµ(t, x, y) ≤ c t−d/α ∧ . (3.24) 1 |x − y|d+α 1 |x − y|d+α  µ d Moreover, d p (t, x, y)dy = 1 for every t ∈ (0, T0] and x ∈ . R R (ii) The function pµ(t, x, y) defined in (i) can be extended uniquely to a positive jointly d d d d continuous function on (0, ∞)×R ×R so that for all s, t ∈ (0, ∞) and (x, y) ∈ R ×R ,  µ d p (t, x, y)dy = 1 and R  pµ(s + t, x, y) = pµ(s, x, z)pµ(t, z, y)dz. (3.25) d R

4. Analysis on perturbed semigroups and resolvents

d For any bounded function f on R , we define  µ µ Pt f (x) := p (t, x, y) f (y)dy. (4.1) d R d d Recall that C∞(R ) is the space of continuous functions on R that vanish at infinity and that ∞ d Cc (R ) is the space of smooth functions with compact supports. Repeating the proofs of [8, Theorem 1] and [9, Proposition 2.4], we get the following result.

∞ d d Theorem 4.1. For any f ∈ Cc (R ) and g ∈ C∞(R ), we have    1 µ α/2 lim (Pt f (x) − f (x))g(x)dx = g(x)∆ f (x)dx + g(x)∇ f (x) · µ(dx). t→0 d d d t R R R The proof of the following result is exactly the same as that of [9, Proposition 2.3].

µ Proposition 4.2. The family of operators {Pt : t ≥ 0} defined above forms a Feller semigroup. Moreover, it satisfies the strong Feller property, that is, for eacht > 0,Pµ maps bounded functions to continuous functions. d It follows from the proposition above that there exists a family {Px : x ∈ R } of probability d d measures on (Ω, F∞) such that (Xt , Px , x ∈ R ) is a conservative Feller process on R such that µ µ,λ Pt f (x) = Ex [ f (Xt )]. For any λ ≥ 0, we will use G to denote the λ-potential density of X:  ∞ Gµ,λ(x, y) = e−λt pµ(t, x, y)dt. 0 n n Recall that Px is the probability measure on (Ω, F∞) induced by the law of X , the α-stable n = µn,λ n process with drift Un, when X0 x, and that G is the λ-potential density of X . 2496 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

n d Theorem 4.3. As n → ∞, Px converges weakly to Px in D([0, ∞), R ) with respect to Skorohod d topology for each x ∈ R . d Proof. It follows from (3.3) that, for any x ∈ R and 0 ≤ t1 < t2 < ··· < tk, the distributions { n} of (Xt1 ,..., Xtk ), Px form a tight sequence. By the same argument as that for [9, (3.1)], we get that n d Px (Xs ̸∈ B(x, r)) ≤ p for all n ≥ 1, 0 ≤ s ≤ t and x ∈ R implies

n  d Px sup |Xs − X0| ≥ 2r ≤ 2p for all n ≥ 1, x ∈ R . s≤t Hence, by (3.3) and the same argument leading to [9, (2.5)], we have for every r > 0,

n  lim sup Px sup |Xs − X0| ≥ 2r = 0. t→0 d s≤t n≥1,x∈R Thus, it follows from the Markov property and [13, Theorem 2] (see also [11, Corollary 3.7.4] d n n and [5, Theorem 3]) that, for each x ∈ R , the laws of {X , Px } form a tight sequence in the d Skorohod space D([0, ∞), R ). Combining this and Theorem 3.9 with [11, Corollary 4.8.7], we d n get that for each x ∈ R , {X, Px } converges to {X, Px } weakly. The following simple result follows immediately from Theorem 3.10. Its proof is the same as that of [9, Lemma 3.1], so we omit its proof.

Lemma 4.4. For any δ > 0,   lim sup Px sup |Xs − x| > δ = 0. t↓0 d s≤t x∈R As a consequence, we immediately get the following corollary. The proof of this corollary is exactly the same as that of [4, Corollary 4.4].

Corollary 4.5. Let β ∈ (0, 1]. There exists δ = δ(d, β) < 1 depending on µ only through the rate at which d Mα (r) goes to zero such that if τ(β) = inf{t : |X − X | > β} ∧ 1, then i=1 µi t 0 supx Ex [exp(−τ(β))] ≤ δ. Fix ν ∈ Kd,α−1. It follows from Lemmas 2.3 and 2.5 and Theorems 3.1 and 3.10 that  1   1  sup pµ(t, x, y)|ν|(dy)dt + sup pUn (t, x, y)|ν|(dy)dt d d d d x∈R 0 R n≥1,x∈R 0 R α ≤ c1 Mν (1), (4.2) where c1, independent of ν and λ ≥ 0, depends on d, α and depends on µ only through the rate at which d Mα (r) goes to zero. Using this and the semigroup property of pµ(t, x, y) and i=1 µi pUn (t, x, y), we get that for any λ ≥ 0 and m = 1, 2,... ,  m+1  e−λt pµ(t, x, y)|ν|(dy)dt d m R  1   = e−λm e−λt pµ(m, x, z)pµ(t, z, y)dz|ν|(dy)dt d d 0 R R P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2497

   1   ≤ e−λm sup e−λt pµ(t, w, y)|ν|(dy)dt pµ(m, x, z)dz d d d R w∈R 0 R −λm α ≤ c1e Mν (1) (4.3) and  m+1  e−λt pUn (t, x, y)|ν|(dy)dt d m R  1   = e−λm e−λt pUn (m, x, z)pUn (t, z, y)dz|ν|(dy)dt d d 0 R R    1   ≤ e−λm sup e−λt pUn (t, w, y)|ν|(dy)dt pUn (m, x, z)dz d d d R n≥1,w∈R 0 R −λm α ≤ c1e Mν (1). (4.4) The assertion of the next lemma now follows immediately.

µ,λ µn,λ Lemma 4.6. For any ν ∈ Kd,α−1, the functions G ν(x) and supn≥1 G ν(x) are bounded −1 α by c(1 + λ )Mν (1), where c > 0 (independent of ν and λ ≥ 0) depends on d, α and depends on µ only through the rate at which d Mα (r) goes to zero. i=1 µi

µ,λ µ ,λ Lemma 4.7. For any λ > 0 and ν ∈ Kd,α−1, as n → ∞,G Vn(x), G n Vn(x) and µ ,λ µ,λ d G n ν(x) all converge to G ν(x), uniformly on compact subsets of R , where Vn(x) :=  ϕn(x − y)ν(dy). µ ,λ µ,λ Proof. We will show that G n Vn(x) converges to G ν(x), uniformly on compact subsets of d R . The other claims are simpler. It follows from (4.2)–(4.4) that   ∞  lim sup e−λt pµ(t, x, y)|ν|(dy)dt T →∞ d d x∈R T R  ∞   −λt Un + sup e p (t, x, y)|Vn(y)|dydt = 0. d d n≥1,x∈R T R By Lemmas 2.3 and 2.5 and Theorems 3.1 and 3.10, we also have   δ  lim sup e−λt pµ(t, x, y)|ν|(dy)dt δ→0 d d x∈R 0 R  δ   −λt Un + sup e p (t, x, y)|Vn(y)|dydt = 0. d d n≥1,x∈R 0 R Thus, it suffices to show that for any0 < δ < T < ∞, the function  T   T  −λt Un −λt µ e p (t, x, y)Vn(y)dydt converges to e p (t, x, y)ν(dy)dt, d d δ R δ R d uniformly on compact subsets of R . Without loss of generality, we only need to prove the uniform convergence on B(0, r) for any r > 0. Fix λ, r > 0 and 0 < δ < T < ∞. It follows 2498 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 from Theorems 3.1 and 3.10 that T pµ t x y ∨ pUn t x y ≤ c t ≤ T ( , , ) (sup ( , , )) 1 d+α , . n |x − y| Thus, by Lemma 3.6, we have that for all large m,  T   T  |ν|(dy) e−λt pµ t x y | | dy dt ≤ c dt ( , , ) ν ( ) 2 d+α δ B(0,mr)c δ B(0,mr)c |y| 1−2α α −α ≤ c3r Mν (r)m . Similarly, using Lemma 2.5 in addition, we have  T  −λt Un 1−2α α −α sup e p (t, x, y)|Vn(y)|dydt ≤ c4r Mν (r)m . n δ B(0,mr)c Therefore,  T  lim e−λt pµ(t, x, y)|ν|(dy)dt →∞ m δ B(0,mr)c  T   −λt Un + sup e p (t, x, y)|Vn(y)|dydt = 0. n δ B(0,mr)c Thus, to complete the proof of this lemma we only need to show that for any m = 1, 2,... ,  T  −λt Un e p (t, x, y)Vn(y)dydt δ B(0,mr)  T  converges to e−λt pµ(t, x, y)ν(dy)dt δ B(0,mr) uniformly on B(0, r). This follows easily from (2.1), Lemmas 2.5 and 3.4 and Theorem 3.9. In fact, by (2.1) and Lemma 2.5,  T   T    −λt Un −λt µ   e p (t, x, y)Vn(y)dydt − e p (t, x, y)ν(dy)dt  δ B(0,mr) δ B(0,mr)   T    −λt Un µ  ≤  e (p (t, x, y) − p (t, x, y))Vn(y)dydt  δ B(0,mr)   T    −λt µ  +  e p (t, x, y)(Vn(y)dy − ν(dy))dt  δ B(0,mr)    d+1−α α Un µ ≤ T (mr) Mν (mr) sup |p (t, x, y) − p (t, x, y)| (t,x,y)∈[δ,T ]×B(0,r)×B(0,mr)    −λt µ  + T sup  e p (t, x, y)(Vn(y)dy − ν(dy)) . (t,x,y)∈[δ,T ]×B(0,r)×B(0,mr)  B(0,mr)  Since pµ(t, x, y) is jointly continuous by Theorem 3.10, applying Lemma 3.4 and Theorem 3.9, we arrive at the last assertion. The following result is similar in spirit to Lemma 4.7. The difference is that the following result gives uniform convergence when the measure ν has compact support. P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2499

µ,λ Lemma 4.8. For any λ > 0 and ν ∈ Kd,α−1 with compact support, as n → ∞,G Vn(x) µ,λ d  converges to G ν(x) uniformly on R , where Vn(x) := ϕn(x − y)ν(dy). For any fixed m, we µ ,λ µ ,λ d also have G m Vn(x) converges to G m ν(x) uniformly on R as n → ∞. Proof. We will prove the first assertion only. The proof of the second assertion is the same.As in the proof of Lemma 4.7,   ∞  lim sup e−λt pµ(t, x, y)|ν|(dy)dt T →∞ d d x∈R T R  ∞   −λt µ + sup sup e p (t, x, y)|Vn(y)|dydt = 0 n d d x∈R T R and   δ  lim sup e−λt pµ(t, x, y)|ν|(dy)dt δ→0 d d x∈R 0 R  δ   −λt µ + sup sup e p (t, x, y)|Vn(y)|dydt = 0. n d d x∈R 0 R Suppose that the support of ν is contained in B(0, R) for some R > 1. Then it follows from Theorem 3.10, (2.1) and Lemma 2.5 that, as m → ∞,  T  sup e−λt pµ(t, x, y)|ν|(dy)dydt c d x∈B(0,m R) δ R  T  −λt µ + sup sup e p (t, x, y)|Vn(y)|dydt → 0. c d n x∈B(0,m R) δ R Thus, it suffices to show that for any m > 0 and 0 < δ < T < ∞, the function  T  −λt µ e p (t, x, y)Vn(y)dydt δ B(0,R+1)  T  converges to e−λt pµ(t, x, y)ν(dy)dt, δ B(0,R+1) uniformly on B(0, m R). This follows easily from Lemma 2.5 and the joint continuity of µ p (t, x, y) (Theorem 3.10).

5. Existence and uniqueness of the stable process with singular drift

In this section we prove Theorem 1.4(a)–(b). The argument of this section is motivated by [4]. The following simple result will play an important role in this section.

d Proposition 5.1. (1) If ν ∈ Kd,α−1 is supported in B(x0, r0) for some x0 ∈ R and r0 > 0, then for all λ ≥ 0, Rλν is a C1 function with λ λ α |∇ R ν(x)| ≤ |∇ R | |ν|(x) ≤ C1 Mν (r0),

where C1 = C1(d, α) > 1 is the constant in (1.5). 2500 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

(2) For all λ > 0, there exists C3 = C3(λ, d, α) > 0 such that for every bounded measurable function g, λ λ |∇ R g(x)| ≤ |∇ R | |g|(x) ≤ C3|g|∞.

Proof. Both assertions are consequences of (1.5). In fact, using (1.5), one can easily check that  λ  λ x → ∇x R (x, y)ν(dy) and x → d ∇x R (x, y)g(y)dy are continuous. Moreover, B(x0,r) R these observations and (1.5) imply that   λ λ λ λ ∇ R ν(x) = ∇x R (x, y)ν(dy), ∇ R g(x) = ∇x R (x, y)g(y)dy. d B(x0,r) R (See the proof of [8, (33)].) Thus, applying (1.5) again, we get   |ν|(dy) |∇ Rλν(x)| ≤ |∇ Rλ(x, y)| |ν|(dy) ≤ C ≤ C Mα(r ) x 1 | − |d+1−α 1 ν 0 B(x0,r0) B(x0,r0) x y and  λ λ |∇ R g(x)| ≤ ∥g∥∞ |∇x R (x, y)|dy d R   α−d−1 −2 −d−α−1 ≤ C1 |y| ∧ (λ |y| )dy ∥g∥∞. d R

In the remainder of this section, for any vector-valued signed measure ν = (ν1, . . . , νd ), we 1 d will use Nν to denote the following operator that maps a C function on R into a measure d  ∂φ N φ(dx) = ∇φ(x) · ν(dx) = (x)ν j (dx). ν ∂x j=1 j

d Recall that α-stable processes with singular drifts on R are defined in Definition 1.3. We will first prove the existence and uniqueness ofthe α-stable process with drift µ under the following additional assumption: d j A: There exist x1 ∈ R and ρ ∈ (0, 1] such that the support of each measure µ is contained in B(x , ρ) and Mα (ρ) < (2dC )−1, where C is the constant in Proposition 5.1. 1 µ j 1 1

Proposition 5.2. Assume that A holds. Suppose that ν ∈ Kd,α−1 is supported in B(x0, r0) for ∈ d α −1 := λ some x0 R and r0 > 0. If Mν (r0) < κ(2dC1) for some κ > 0, then ν Nµ R ν and := λ + −n νn Nµn R ν are in Kd,α−1 and are supported in B(x1, ρ) and B(x1, ρ 2 ) respectively. Moreover, for all r > 0,

d d κ  κ  Mα(r) < Mα (r) and Mα (r) < Mα (r). (5.1) ν 2d µ j νn 2d µ j j=1 j=1 In fact, we have

d d  κ  |ν|(dx) ≤ |∇ Rλν(x)| |µ j |(dx) ≤ |µ j |(dx) (5.2)  2d j=1 j=1 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2501 and d d  λ j κ  j |νn|(dx) ≤ |∇ R νn(x)| |µ |(dx) ≤ |µ |(dx). (5.3)  n 2d n j=1 j=1

Proof. Proposition 5.1 implies (5.2). In fact, d d d   κ  |∇ Rλν(x)| |µ j |(dx) ≤ C Mα(r ) |µ j |(dx) ≤ |µ j |(dx), 1 ν 0 2d j=1 j=1 j=1 so the first part of (5.1) follows immediately. Similarly, Proposition 5.1 implies (5.3) so d d  j κ  j |∇ Rλν(x)| |µ |(dx) ≤ |µ |(dx). n 2d n j=1 j=1 Now using Lemma 2.5, the second part of (5.1) follows. µ ,λ n Recall that G n is the λ-potential density of X , the α-stable process with drift Un.

Lemma 5.3. Suppose that A holds. If g is bounded and ν ∈ Kd,α−1 has compact support, then for any λ > 0,

∞  ∞  µn,λ  λ λ k λ  λ k G g(x) = R (Nµn R ) g(x) = R (Nµn R ) g(x) (5.4) k=0 k=0 and ∞  ∞  µn,λ  λ λ k λ  λ k G ν(x) = R (Nµn R ) ν(x) = R (Nµn R ) ν(x). (5.5) k=0 k=0 ∞ λ λ k ∈ d Furthermore, the series k=0 R (Nµn R ) ν(x) converges uniformly in x R and the rate of α α convergence depends on µ and ν only via the rates at which Mµ(r) and Mν (r) tend to zero. Proof. If f ∈ C2 is bounded along with its first and second order partial derivatives, then by Ito’s formula (see, for instance, [1, Theorem 4.4.7]), d  t n  n i n f (Xt ) − f (X0) = ∂i f (Xs−)Un(Xs )ds i=1 0  t  n n + [ f (Xs− + x) − f (Xs−)]N(ds, dx) 0 |x|<1  t  n n + [ f (Xs− + x) − f (Xs−)]N(ds, dx) 0 |x|≥1  t  n n n + [ f (Xs− + x) − f (Xs−) − x · ∇ f (Xs−)]J(x)dxds, 0 |x|<1 where J(x) is the Levy´ density of S, i.e., d + α   α −1 J(x) := α2α−1π −d/2Γ Γ 1 − |x|−d−α, (5.6) 2 2 2502 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

N is the Poisson random measure describing the jumps of the symmetric stable process S and

N(ds, dx) := N(ds, dx) − ds J(x)dx. (5.7) n −λt Taking Px expectations, multiplying by e and integrating over t from 0 to ∞, we obtain

µn,λ µn,λ α/2 λG f (x) = f (x) + G [∇ f · Un + ∆ f ](x). (5.8)

We first consider ν ∈ Kd,α−1 under the extra assumption that the support of ν is contained in α −1  B(x0, r0) for some r0 > 0 and Mν (r0) < (2dC1) . Let Vm(x) := ϕm(x−y)ν(dy), which is in ∞ d α/2 λ λ λ Cc (R ) and satisfies ∆ R Vm = λR Vm −Vm. Thus, with f = R Vm, we get from (5.8) that

µn,λ λ µn,λ λ G Vm(x) = R Vm(x) + G Nµn R Vm(x). (5.9) µ ,λ µ ,λ λ λ By Lemma 4.8, G n Vm(x) → G n ν(x) and R Vm(x) → R ν(x). Moreover, by Lemma 3.5, λ λ ∇ R Vm converges uniformly to ∇ R ν. Thus, combining this with Lemma 4.6, we have that  d ∂ µn,λ λ µn,λ  λ j G Nµn R Vm(x) = G (x, y) R Vm(y)µn(dy) d ∂x R j=1 j converges to  d ∂ µn,λ λ µn,λ  λ j G Nµn R ν(x) = G (x, y) R ν(y)µn(dy) d ∂x R j=1 j as m → ∞. Therefore, by (5.9),

µn,λ λ µn,λ λ G ν(x) = R ν(x) + G Nµn R ν(x). λ The finiteness is guaranteed by Proposition 5.1. Since Proposition 5.2 implies that Nµn R ν ∈ λ + −n Kd,α−1 and the support of Nµn R ν is contained in B(x1, ρ 2 ), we can iterate the last relation and get

µn,λ λ λ λ µn,λ λ λ G ν(x) = R ν(x) + R Nµn R ν(x) + G Nµn R Nµn R ν(x). Continuing this iteration we get m µn,λ  λ λ k µn,λ λ m+1 G ν(x) = R (Nµn R ) ν(x) + G (Nµn R ) ν(x). (5.10) k=0 j Repeatedly applying (5.3) with κ = 1, first to ν and then to µn, we get  µn,λ λ m+1 µn,λ λ m+1 |G (Nµn R ) ν(x)| ≤ G (x, y)|(Nµn R ) ν|(dx) d R d −m  µn,λ j ≤ c12 G |µn|(x). j=1

µn,λ λ m Thus, by Lemma 4.6, G (Nµn R ) ν(x) → 0 uniformly in x as m → ∞. Similarly, we have ∞ λ λ k → → ∞ k=m+1 R (Nµn R ) ν(x) 0 as m . Therefore ∞ µn,λ  λ λ k G ν(x) = R (Nµn R ) ν(x). k=0 This establishes (5.5) under the extra assumption above. P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2503

If ν ∈ Kd,α−1 has compact support, then it can be written as the sum of finitely many measures in ν ∈ Kd,α−1 satisfying the extra assumption above. Thus, by linearity, (5.5) holds for general ν ∈ Kd,α−1 with compact support. By setting ν(dx) = g(x)dx, we get that (5.4) holds for bounded functions with compact support. λ λ Finally, for g bounded, let gm(x) = g(x)1{|x|≤m}. Clearly, R gm → R g. Note that, under j −n assumption A, µn has support in the ball B(x1, ρ + 2 ) for each j. Using (1.5), we see that λ λ −n ∇ R gm → ∇ R g uniformly and boundedly in B(x1, ρ + 2 ) so that λ λ λ λ |R (Nµn R )gm(x) − R (Nµn R )g(x)| d     λ  ∂ λ ∂ λ  j ≤ R (x, y)  R gm(y) − R g(y) |µn|(dy) ∂x ∂x j=1 B(x1,ρ)  j j  −n and this converges uniformly and boundedly in B(x1, ρ + 2 ) to 0 as m → ∞. Inductively we λ λ k λ λ k see that R (Nµn R ) gm(x) → R (Nµn R ) g(x) as m → ∞. λ λ Moreover, Nµn R gm(dx) and Nµn R g(dx) are supported in the ball B(x1, ρ) and by Propo- sition 5.1(2), for every m ≥ 1,

d d α α  α  α M λ (ρ) ∨ M λ (ρ) ≤ c∥g∥∞ M j (ρ) ≤ c2∥g∥∞ M j (ρ) Nµn R gm Nµn R g µn µ j=1 j=1 ≤ ∥ ∥ −1 −1 c2 g ∞2 C1 . Thus, by Proposition 5.2,

d λ 2 λ 2 −1  j |(Nµn R ) gm(dx)| ∨ |(Nµn R ) g(dx)| ≤ c2∥g∥∞2 |µn|(dx) j=1 and d α α −1  α −2 −1 M λ 2 (ρ) ∨ M λ 2 (ρ) ≤ c2∥g∥∞2 M j (ρ) ≤ c2∥g∥∞2 C . (Nµn R ) gm (Nµn R ) g µ 1 j=1 Thus, inductively,

Lemma 5.4. Suppose that A holds. If λ > 0, then

µm ,λ j j lim sup |G (µn − µ )(x)| = 0. n→∞ d x∈R ,m≥1

Proof. It follows from (4.2)–(4.4), Lemmas 2.3, 2.5 and Theorems 3.1, 3.10 that   ∞   −λt Um lim sup sup e p (t, x, y)(|µ|(dy) + |µn|)(dy)dt = 0 T →∞ n,m d d x∈R T R and   δ   −λt Um lim sup sup e p (t, x, y)(|µ|(dy) + |µn|)(dy)dt = 0. δ→0 n,m d d x∈R 0 R

Since the support of |µ| + |µn| is contained in B(x1, ρ + 1) ⊂ B(0, R) where R := ρ + 1 + |x1|, it follows from Theorem 3.10, (2.1) and Lemma 2.5  T   −λt Um lim sup sup e p (t, x, y)(|µ|(dy) + |µn|)(dy)dt = 0. k→∞ c d n,m x∈B(0,k R) δ R Thus, it suffices to show that for every k ≥ 1,

−l0 α Given ε > 0 we choose l0 > 0 large so that c22 Mµ(1) < ε, then by (5.10) and (5.17),

l0  λ λ l ≤ sup |R (Nµm R ) (µn − µ)(x)| + ε. l=0 x∈B(0,k R),m≥1 Now by applying (5.16), we have proved (5.13). Recall that Gµ,λ is the λ-potential density of X.

Proposition 5.5. Suppose that A holds. If λ > 0 and g is bounded, then  ∞  µ,λ λ  λ k G g = R (Nµ R ) g. k=0

∞ µn,λ = λ  λ k Proof. Let g be bounded. By Lemma 5.3, we have G g R ( k=0(Nµn R ) )g, and, by Lemma 4.7, we have Gµn,λg → Gµ,λg as n → ∞. In view of A, Propositions 5.1 and 5.2 (see (5.12) for a similar computation), we have d λ λ k λ λ k −k+1  λ j ∥R (Nµ R ) g∥∞ ∨ ∥R (Nµn R ) g∥∞ ≤ c12 ∥g∥∞ ∥R |µn|∥∞. j=1 Thus by Lemmas 2.5 and 4.6, d λ λ k λ λ k −k+1  ∥R (Nµ R ) g∥∞ ∨ ∥R (Nµn R ) g∥∞ ≤ c22 ∥g∥∞ Mµ(1). j=1 For k ≥ 1, write k−1 λ λ k λ λ k  λ λ l λ λ λ k−l−1 R (Nµn R ) g − R (Nµ R ) g = R (Nµn R ) (Nµn R − Nµ R )(Nµ R ) g. l=0 2506 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

λ m By Propositions 5.1 and 5.2, the measure νm defined by νm(g) = (Nµ R ) g (and ν0(dx) = g(x)dx) is in Kd,α−1 and for m ≥ 1, νm has support in the ball B(x1, ρ) and its total variation is −m d j bounded by 2 (κ/d)∥g∥∞ j=1 |µ | for some κ > 0 (see (5.11) for a similar computation). By Proposition 5.2, for 0 ≤ l ≤ k − 1, the measure defined by d λ λ λ k−l−1  ∂ λ j j (N R − N R )(N R ) g(dx) = R (νk−l− (g))(x)(µ − µ )(dx) µn µ µ ∂x 1 n j=1 j is in Kd,α−1. Since, by Propositions 5.1 and 5.2, d d  ∂ λ j j α  j j | R (νk−l−1(g))(x)(µn − µ )(dx)| ≤ C1 M (ρ) |µn − µ |(dx) ∂x νk−l−1(g) j=1 j j=1 d −k+l  j j ≤ c2∥g∥∞2 |µn − µ |(dx), j=1 the measure defined by λ λ λ λ k−l−1 (Nµn R )(Nµn R − Nµ R )(Nµ R ) g −k+l d j is in Kd,α−1 and its total variation is bounded by c3∥g∥∞2 βn j=1 |µn|(dx), where d     λ j j  βn = sup  ∇x R (x, y)(µn − µ )(dy) , j=1 x  B(x1,ρ)  which goes to zero as n → ∞ by Lemma 3.5. By Proposition 5.1, when 1 ≤ l ≤ k − 1 the measure defined by λ l λ λ λ k−l−1 (Nµn R ) (Nµn R − Nµ R )(Nµ R ) g −k d j is in Kd,α−1 and its total variation is bounded by c4∥g∥∞2 βn j=1 |µn|(dx), thus, by λ λ l λ λ λ k−l−1 Lemma 2.5, R (Nµn R ) (Nµn R − Nµ R )(Nµ R ) g is a function whose sup norm is less −k d d α than or equal to c ∥g∥∞2  β  M (ρ). This implies that this term goes to zero as 5 j=1 n j=1 µ j n → ∞. d n d Recall that {Px : x ∈ R } and {Px : x ∈ R } are the probability measures on (Ω, F∞) = n n = induced by the laws of X with X0 x and X with X0 x respectively. d The following result says that, under assumption A, for each x ∈ R , Px is a solution of (1.8) such that Px (X0 = x) = 1.

d Proposition 5.6. Suppose that A holds. For each x ∈ R , under Px we have Xt = x + St + At d and Px (X0 = x) = 1, where St is a symmetric α-stable process on R starting from the origin =  t d and At limn→∞ 0 Un(Xs)ds is an R -valued continuous additive functional of X having finite variation, with the convergence in the following sense: forany ϵ > 0 and t > 0,   s  lim sup Px sup | Un(Xr )dr − As| > ϵ = 0. n→∞ d s≤t x∈R 0 { } ∈ d  t | | Furthermore, there exists a subsequence nk such that for every x R , supk 0 Unk (Xs) ds < ∞Px -a.s. for each t > 0. P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2507  Proof. For any positive measure ν ∈ Kd,α−1, since Vn(x) := ϕn(x − y)ν(dy) is a bounded n function and (Xt , Px ) satisfies the strong Feller property, we know that foreach n ≥ 1 and µ ,λ µ,λ j = 1,..., d, G n Vn is continuous. Thus, by Lemmas 4.6 and 4.7, G ν is a bounded continuous potential of X. For any j = 1,..., d, we use µ j,+ and µ j,− to denote the positive j j,+ j,− and negative parts of µ respectively. We will use µn and µn to denote the approximations j,+ j,− µ,λ j,+ µ,λ j,+ of µ and µ respectively. Then it follows from Lemma 4.8 that G µn → G µ µ,λ j,− µ,λ j,− d µ,λ j,+ µ,λ j,− and G µn → G µ uniformly in x ∈ R . Hence, G µ and G µ are bounded continuous potentials of X. Therefore, by [7, Theorem IV.3.13], there are positive j,+ j,− µ,λ j,+ =  ∞ −λt j,+ continuous additive functionals A and A of X such that G µ Ex 0 e d At µ,λ j,− =  ∞ −λt j,− j = j,+ − j,− = 1 d and G µ Ex 0 e d At . Let A A A and A (A ,..., A ). By Lemma 4.8 we know that  µ,λ j j  lim sup G (µn − µ )(x) = 0. n→∞ d x∈R 2 n It follows from [4, Lemma 3.10] that lim →∞ sup d [sup |B | ] = 0, where B = n x∈R Ex t t (B1,n,..., Bd,n) and  t  t j,n −λs j −λs j Bt = e d As − e Un (Xs)ds. 0 0 j,n = j −  t j n = 1,n d,n If Ht At 0 Un (Xs)ds and H (H ,..., H ), then using integration by parts we get  t  t j,n λs j,n λt j,n λs j,n Ht = e d Bs = e Bt − λe Bs ds. (5.18) 0 0 Therefore,   s     2 n 2 lim sup Ex supAs − Un(Xr )dr = lim sup Ex sup |Hs | = 0. (5.19) n→∞ d s≤t n→∞ d s≤t x∈R 0 x∈R  t j j Hence, 0 Un (Xs)ds converges to At uniformly in t over finite intervals in probability. There- { }  t j j fore, there exists a subsequence nk such that 0 Unk (Xs)ds converges to At a.s. for each j, uniformly in t over finite intervals. Note that our subsequence {nk} has been chosen independent of x so that we have this convergence Px -a.s. for each x. j j,+ j,− j j If we let νn = µn + µn , then νn converges weakly to |µ |. By the same argument as in  t j the previous paragraph, by choosing further subsequence if necessary, 0 νnk (Xs)ds converges j j uniformly in t over finite intervals a.s. As |Un (x)| ≤ νn (x) for each j, n and x, it follows that  t | j | ∞ supk 0 Unk (Xs) ds < a.s. By Lemma 5.3, we know that  ∞  µm ,λ j λ  λ k j G |µn| = R (Nµm R ) |µn|(x) k=0 −1 α and, by Lemmas 2.5 and 4.6, it is bounded independently of n and m by c1(1 + λ )Mµ(1). Hence, by Lemma 5.4 and [4, Lemma 3.10],    t  t 2 m  −λs j −λs j  sup Ex sup e Up (Xs)ds − e Un (Xs)ds → 0 d ≥   x∈R ,m≥1 t 0 0 0 2508 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 as n, p → ∞ and then   s  s 2 m j j lim sup Ex sup Up (Xr )dr − Un (Xr )dr = 0. n,p→∞ d s≤t x∈R ,m≥1 0 0 Therefore, there exists a further subsequence {n } of {n } such that  t U j (X )ds converges k k 0 nk s m  Px -a.s. for each m, uniformly over t in finite intervals and the rate of convergence is independent d of m. Therefore, for any fixed ξ ∈ R , t > 0 and ε > 0 there exists k0 ≥ 1 such that for every k, l ≥ k0,

  t   t    iξ· X − U (X )ds   iξ· X − U (X )ds   m t 0 nk s m t 0 nl s  sup Ex e − Ex e  < ε. (5.20) d   x∈R ,m≥1  t j j By (5.19), we also have that U (Xs)ds converges to At x -a.s., uniformly over t in finite 0 nk P d intervals. Therefore, for any ξ ∈ R , t > 0 and ε > 0 there exists k1 ≥ k0 such that for every k ≥ k1,     · − t     iξ Xt 0 Unk (Xs )ds iξ·(Xt −At )  lim sup Ex e  − Ex e  < ε. (5.21) k→∞ d   x∈R j − t j For any fixed n and t, by using [14, Theorem 15.12], we know that ω (t) 0 Un (ω(s))ds is a d continuous function on the Skorohod space D([0, ∞), R ), therefore by the weak convergence,   t    t  m iξ·(Xt − Un(Xs )ds) iξ·(Xt − Un(Xs )ds) d lim e 0 = x e 0 for all ξ, x ∈ . (5.22) m→∞ Ex E R x ∈ d t n = n m = n For given ξ, R , > 0 and ε > 0, take l k1 and let k along the subsequence {nk}. Then, combining (5.20)–(5.22), if k is large enough,   t    iξ· X − U (X )ds     nk t 0 nk s iξ·(Xt −At )  Ex e − Ex e  ≤ 3ϵ.   That is,

 · − t   · −  nk iξ (Xt 0 Unk (Xs )ds) iξ (Xt At ) lim Ex e  = Ex e . k→∞ −  t − Since Un is a bounded and smooth function, Xt 0 Un(Xs)ds x is a symmetric α-stable n process starting from 0 under Px so that  · − t −  − | |α nk iξ (Xt 0 Unk (Xs )ds x) t ξ Ex e  = e .

iξ·(X −A −x) −t|ξ|α Thus Ex [e t t ] = e . Consequently Xt − At − x is a symmetric α-stable process on d R under Px .

Remark 5.7. It follows from the proof above that, given a subsequence {nk}, there exists a { } ∈ d further subsequence nkm such that for every x R ,   t  | | ∞ = Px sup Unkm (Xs) ds < 1 m≥1 0 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2509 and for every T > 0,    t  lim sup  U (X )ds − A  = 0 = 1. Px →∞  nkm s t  m 0≤t≤T 0

We now show that, under assumption A, (1.8) has a unique solution.

Proposition 5.8. Assume that A holds. If Px is a solution to (1.8), then Px = Px . Proof. Using our Remark 5.7, by following the argument in [4, pp. 812–813] line by line, in proving the uniqueness, it suffices to consider solutions Px for which there exists a subsequence {nk} with d  ∞ d  ∞  −λt j  −λt j e d|A |t < ∞ and sup e |Un (Xt )|dt < ∞. (5.23) EPx EPx k j=1 0 k j=1 0 Thus throughout the proof, we assume that (5.23) holds. 2 Let Px be such a solution and let f ∈ C be bounded along with its first and second order partial derivatives. By Ito’s formula (see, for instance, [1, Theorem 4.4.7]),

d  t  µ i f (Xt ) − f (X0) = ∂i f (Xs−)d As i=1 0  t  µ µ + [ f (Xs− + x) − f (Xs−)]N(ds, dx) 0 |x|<1  t  µ µ + [ f (Xs− + x) − f (Xs−)]N(ds, dx) 0 |x|≥1  t  µ µ µ + [ f (Xs− + x) − f (Xs−) − x · ∇ f (Xs−)]J(x)dxds, 0 |x|<1 where N is the Poisson random measure describing the jumps of the symmetric stable process S and J and N are defined in (5.6) and (5.7). Taking expectations with respect to Px , multiplying by e−λt and integrating over t from 0 to ∞, we get  ∞ 1 e−λt f (X )dt − f (x) Ex t P 0 λ 1  ∞ 1  ∞ = e−λt ∇ f (X ) · d A + e−λt ∆α/2 f (X )dt. Ex t t Ex t λ P 0 λ P 0 Suppose g ∈ C2 is bounded and set f = Rλg, then f ∈ C2 is bounded along with its first and second order partial derivatives and satisfies ∆α/2 f = λRλg − g. Substituting, we get  ∞  ∞ e−λt g(X )dt = Rλg(x) + e−λt ∇ Rλg(X ) · d A . (5.24) Ex t Ex t t P 0 P 0 Define  ∞ V λh = e−λt h(X )dt. Ex t P 0 2510 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

We can then rewrite (5.24) as  ∞ V λg = Rλg(x) + e−λt ∇ Rλg(X ) · d A . (5.25) Ex t t P 0 By taking limits, we can show that (5.25) holds for all bounded continuous functions g.  t Since 0 Unk (Xs)ds converges to At uniformly over t in finite intervals, by using (5.23) we  ∞ ·  ∞ · → get that 0 Hs Unk (Xs)ds converges to 0 Hs d As, when s Hs is a simple function. By approximating cadlag functions with simple functions, we again have convergence, when s → Hs is cadlag. Therefore the second term on the right-hand side of (5.25) is equal to  ∞ lim e−λt ∇ Rλg(X ) · U (X )dt = lim V λ((∇ Rλg) · U ). Ex t nk t nk k→∞ P 0 k→∞ Applying (5.25) again, we get λ λ λ λ V ((∇ R g) · Unk ) = R ((∇ R g) · Unk )(x)  ∞ + e−λt ∇ Rλ((∇ Rλg) · U )(X ) · d A . (5.26) Ex nk t t P 0 λ λ The limit as k → ∞ of the first term on the right-hand side of (5.26) is R (Nµ R g)(x). For the second term on the right-hand side of (5.26), the integral is dominated by  ∞ e−λt |∇ Rλ(N Rλg)(X )|d|A| , µnk t t 0 and |∇ Rλ(N Rλg)| is uniformly bounded by Proposition 5.1. Therefore the limit of the second µnk term of (5.26) is  ∞ e−λt ∇ Rλ(N Rλg)(X ) · d A . Ex µ t t P 0 We thus have  ∞ V λg = λg(x) + Rλ N Rλg(x) + e−λt ∇ Rλ(N Rλg)(X ) · d A . R µ Ex µ t t P 0 We continue by writing the last expression as the limit of  ∞ e−λt ∇ Rλ(N Rλg)(X ) · U (X )dt = V λ(∇ Rλ(N Rλg) · U ). Ex µ t nk t µ nk P 0 After k steps we arrive at  k   ∞ λ λ  λ l −λt λ λ k V g = (Nµ R ) g(x) + e ∇ R (Nµ R ) g(Xt ) · d At . R EPx l=0 0 −k  ∞ The absolute value of the last term is bounded by c1∥g∥∞2 d|A|t , which tends to 0 as EPx 0 k → ∞. Since     ∞  ∞  λ  λ l  ≤ ∥ ∥  −l → R (Nµ R ) g(x)  c2 g ∞ 2 0  l=k+1  l=k+1 as k → ∞, we have  ∞  λ λ  λ l µ,λ V g = R (Nµ R ) g(x) = G g(x). l=0 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2511

By the uniqueness of Laplace transforms, we have [g(Xt )] = [g(Xt )] for all t, that is, EPx EPx the distributions of Xt under Px and Px are the same. Using this, our Proposition A.1 below and the argument of [2, Theorem VI.3.2], we can easily show the finite-dimensional distributions of Px and Px are the same, we omit the details. Thus Px = Px . Now we are ready to remove assumption A to get the existence and uniqueness of the solution to (1.8). Proof of Theorem 1.4. We can find ρ > 0 such that d Mα (ρ) < (2dC )−1, where C is j=1 µ j 1 1 the constant in Proposition 5.1. Let T0 = 0 and Tk+1 = inf{t > Tk : |Xt − XTk | ≥ ρ} ∧ (Tk + 1) 1 for k ≥ 1. Let Px be the solution to (1.8) with X0 = x and µ replaced by µ|B(x,ρ). Let Qx = Px and define inductively for k ≥ 0 k+1 (F ∩ (B ◦ θT )) = k [ X (B); F], F ∈ FT , B ∈ F∞, Qx k EQx P Tk k where, for each t ≥ 0, θt is the shift operator on Ω defined by (θt ω)(·) = ω(t + ·). Clearly, m| = k | if m ≥ k. Define Qx FTk Qx FTk = k ∈ Px (A) Qx (A), A FTk . Note that

−Tk+1 −(Tk+1−Tk ) −Tk −Tk −T1 k+1 [e ] = k+1 [e e ] = k [e [e ]]. (5.27) E x E x E x EX Q Q Q P Tk Since, by Corollary 4.5, sup [e−T1 ] ≤ δ < 1, by induction the left-hand side of (5.27) is x EPx k+1 [ −Tk ] ≤ k → → ∞ less than or equal to δ . So EPx e δ 0, which implies that Tk , Px -a.s for all d x ∈ R . d We fixt > 0 and x ∈ R . By Proposition 5.6, for every ε > 0,    t∧T1   −  lim sup Py sup  Un(Xs)ds At∧T1  > ϵ n→∞ d   y∈R t≤t 0    t∧T1  =  −  = lim sup Py sup  Un(Xs)ds At∧T1  > ϵ 0. n→∞ d   y∈R t≤t 0 Thus, by the definition of Px ,    t∧Tk+1   − −  lim Px sup  Un(Xs)ds (At∧Tk+1 At∧Tk ) > ϵ n→∞  ∧  t≤t t Tk     t∧T1 = lim k X sup | Un(Xs)ds − At∧T | > ϵ n→∞ EQx P Tk 1 t≤t 0    t∧T1  ≤  −  = lim sup Py sup  Un(Xs)ds At∧T1  > ϵ 0. (5.28) n→∞ d   y∈R t≤t 0 ∧ Therefore,  t Tk+1 U j (X )ds → A j − A j in probability with respect to uniformly in t∧Tk n s t∧Tk+1 t∧Tk Px ∈ [ ]  t j → j ∈ [ ] t 0,t for each k, which implies that 0 Un (Xs)ds At under Px uniformly in t 0,t . Similarly, by Proposition 5.6, there exists a subsequence {nm} such that    t∧T1 inf y sup |Unm (Xs)|ds < ∞ = 1. d P y∈R m≥1 0 2512 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

  t  | | ∞ = Px sup Unm (Xs) ds < 1. (5.29) m≥1 0

Finally, to prove that Px is a solution to (1.8) with X0 = x, it suffices to show that d St := Xt − x − At is a symmetric α-stable process with Px (St = 0) = 1. Fix ξ ∈ R and let  α Mt := exp iξ · St + t|ξ| . ∈ d Note that Mt∧T1 is a Py-martingale for every y R . Using this and the definition of Px , by induction,  α x [Mt∧T ] = k+1 [Mt∧T ] = k+1 [exp iξ · St∧T + (t ∧ Tk+1)|ξ| ] E k+1 EQx k+1 EQx k+1  α  α = k+1 [exp iξ · St∧T ◦ θT + (t ∧ (Tk+1 − Tk))|ξ| exp (t ∧ Tk)|ξ| ] EQx 1 k α α = k [exp((t ∧ T )|ξ| ) [exp(iξ · S ∧ + (t ∧ T )|ξ| )]] E x k EX t T1 1 Q P Tk α = k [exp((t ∧ Tk)|ξ| )] = x [Mt∧T ] = · · · = x [Mt∧T ] = 1. EQx E k E 1 d α By letting k → ∞ we have that for every ξ ∈ R Ex [exp (iξ · St + t|ξ| )] = 1. Therefore St is a symmetric α-stable process with Px (St = 0) = 1. We have proved the existence. Using the arguments similar to those in [2, Proposition VI.2.1, Lemma VI.3.3 and Theo- rem VI.3.4], one can easily prove the uniqueness. We provide the details in the Appendix. Now we prove part (b). The strong Markov property follows from the uniqueness in (a), so we only need to prove the assertion about the continuous additive functional At . We fix t > 0 below. j,n :=  · j j [ ] Since A· 0 Un (Xs)ds converges to A· uniformly on 0, t in probability and (5.29) holds, there exist a subsequence {nm} and an event Ω of full probability such that, for every ω ∈ Ω,  · j j [ ] 0 Unm (Xs(ω))ds converges to A· (ω) uniformly on 0, t and  t sup |Unm (Xs(ω))|ds < ∞. (5.30) m≥1 0 ∈  t · j  t j Therefore, for every ω Ω, 0 Hs Unm (Xs(ω))ds converges to 0 Hs(ω)d As (ω) when s → Hs(ω) is a cadl` ag` step function (with a finite number of jumps). By approximating cadl` ag` functions with cadl` ag` step functions uniformly on [0, t] (see [14, Theorem 15.5]) and ∈  t · j using (5.30), we can show that, for every ω Ω, 0 Hs(ω) Unm (Xs(ω))ds converges to  t j → 0 Hs(ω)d As (ω) when s Hs(ω) is a bounded cadl` ag` function. P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2513

d  t j In particular, when f is a bounded continuous function on R , 0 f (Xs)Unm (Xs)ds con-  t j verges a.s. to 0 f (Xs)d As . Recall that for any positive continuous additive functional Bt of X, 2 =  t − (Bt ) 2 0 (Bt Bs)d Bs, thus by the Markov property we have 2 2 Ex [(Bt ) ] ≤ 2 sup (Ey Bt ) . d y∈R Combining this with Lemma 2.3 and (3.3), we get that, for any fixed t > 0, and every bounded Borel measurable function f  t 2 j sup Ex Un (Xs) f (Xs)ds < ∞, n 0 { t j : ≥ } which implies that the family 0 Un (Xs) f (Xs)ds n 1 is uniformly integrable under Px . d  t j Since, when f is a bounded continuous function on R , 0 Unm (Xs) f (Xs)ds converges to  t j 0 f (Xs)d As a.s., applying Lemmas 2.1, 2.3 and 3.4, we get  t  t  j µ j Ex f (Xs)d As = p (s, x, y) f (y)µ (dy)ds. d 0 0 R Applying the monotone class theorem, we can easily show that the above is true for any bounded d Borel function on R . Part (c) follows from Theorem 3.10. Acknowledgments

The authors thank the referee for many helpful comments on the first version of this paper. Panki Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2013004822). Renming Song’s research was supported in part by a grant from the Simons Foundation (208236).

Appendix. Some details on the uniqueness

d Suppose that P is the law of an α-stable process with drift µ ∈ Kd,α−1 started at x ∈ R , see Definition 1.3. Recall that Ft = σ{Xs : s ≤ t}. Suppose that S is a bounded stopping time with d respect to {Ft }. Define a probability measure PS on Ω = D([0, ∞), R ) by

PS(A) := P(A ◦ θS), A ∈ F∞. (A.1) ′ Here θS is the shift operator that shifts the path by S. Let QS(ω, dω ) be a regular conditional d probability for PS( · |FS). Since D([0, ∞), R ) equipped with the Skorohod topology is ′ a complete separable metric space, such regular conditional probability PS(ω, dω ) exists ′ uniquely. For the existence and uniqueness of regular conditional probability PS(ω, dω ), see, for instance, [15, Theorem 1.3.1]. Using an argument similar to that in the proof of [2, Proposition VI.2.1], it is easy to check d that, with probability one, QS(ω, ·) is an α-stable process with drift µ on R starting from X S(ω). Here we provide the details.

Proposition A.1. With probability one, QS(ω, ·) is the law of an α-stable process with drift µ started at X S(ω). 2514 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

′ ′ Proof. By the same argument as that in the proof of [2, Proposition VI.2.1], QS(ω, {ω : X0(ω ) = X S(ω)}) = 1 for P-a.s. Therefore, to prove the proposition, it suffices to show that Xt −X0−At is a symmetric α-stable process starting from the origin under QS(ω, ·) with probability one. For d this we only need to show that, for any ξ ∈ R , α Mt = Mt (ξ) := exp(iξ · (Xt − X0 − At ) + t|ξ| ) d is a martingale for every ξ ∈ R under QS(ω, ·) with probability one. Let u > t. Since Mt ◦θS = Mt+S/MS is a martingale with respect to FS+t , [ ◦ ; ◦ ∩ ] = [ ◦ ; ◦ ∩ ] EP Mu θS F θS B EP Mt θS F θS B whenever F ∈ Ft and B ∈ FS. This is the same as saying [ ◦ ; ] = [ ◦ ; ] EP (Mu1F ) θS B EP (Mt 1F ) θS B . Since the above holds for all B ∈ FS, by the definition of QS, for each F ∈ Ft , [ ; ] = [ ; ] EQS(ω,·) Mu F EQS(ω,·) Mt F P-a.s.

Being the Borel σ-field of a complete separable metric space, Ft is countably determined. Thus the null set in the display above can be chosen to be independent of F. Since Mt is right continuous in t, the null set in the display above can be chosen to be independent of t. Thus, Mt is a martingale under QS(ω, ·) with probability one. Furthermore, since ξ → Mt (ξ) is continuous, d by considering rational coordinate ξ, we conclude that Mt (ξ) is a martingale for every ξ ∈ R under QS(ω, ·) with probability one. The following result is similar to [2, Lemma VI.3.3].

Lemma A.2. Let S = inf{t > 0 : |Xt − X0| ≥ r} ∧ 1. Suppose that P1 and P2 are the laws of d α-stable processes with drifts µ1 ∈ Kd,α−1 and µ2 ∈ Kd,α−1, respectively, started at x ∈ R . · [ · | ] = ◦ Let Q(ω, ) be a regular conditional probability for EP2,S FS , where EP2,S (B) P2(B θS). Define P by ◦ ∩ = [ ; ] ∈ ∈ P(F θS B) EP1 Q(F) B , B FS, F F∞.

If µ1 and µ2 agree on B(x, r), then P is the law of an α-stable process with drift µ2.

Proof. It is clear that the restriction of P to FS is equal to the restriction of P1 to FS. Hence

P(X0 = x) = P1(X0 = x) = 1. d For any ξ ∈ R , the process α Mt := exp(iξ · (Xt∧S − X0 − A1,t∧S) + (t ∧ S)|ξ| ) α = exp(iξ · (Xt∧S − X0 − A2,t∧S) + (t ∧ S)|ξ| ) is a martingale under P1, where A1 and A2 stand for the continuous additive functionals associated to µ1 and µ2 respectively. Since for each t these random variables are FS-measurable, Mt is also a martingale under P. In fact, for any 0 < s < t,

P(Mt |Fs) = P(Mt |Fs∧S) = P1(Mt |Fs∧S) = P1(Mt |Fs) = Ms. α It remains to show that Nt := exp(iξ · (X S+t − X S − A2,S+t + A2,S) + t|ξ| ) is a martingale under P. This follows immediately from Proposition A.1 and the definition of P. P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2515

The following result is similar to [2, Theorem VI.3.4].

d Theorem A.3. Suppose that µ ∈ Kd,α−1. Then for any x ∈ R , the α-stable process with drift µ started from x is unique. d Proof. By the definition of Kd,α−1, there exists ρ > 0 such that, for any x ∈ R , the restriction µ|B(x,ρ) of µ to B(x, ρ) satisfies condition A. Thus, by Proposition 5.8, for any starting point, the α-stable process with drift µ|B(x,ρ) is unique. d Fix x ∈ R and suppose that P1 and P2 are the laws of two α-stable processes with drift µ started at x. Define T0 = 0 and Tk+1 = inf{t > Tk : |Xt − XTk | ≥ ρ} ∧ (Tk + 1) for k ≥ 1. For simplicity, we will sometimes abbreviate T1 as T . 1 | 1 Write P for the law of the α-stable process with drift µ B(x,ρ) started at x. Let QT be the regular conditional probability defined in (A.1) using P1 and T instead of P and S. For l = 1, 2, define ◦ ∩ = [ 1 ; ] ∈ ∈ Pl (F θT B) EPl QT (F) B , B FT , F F∞. (A.2) We now show that the laws P1 and P2 of any two α-stable processes with drift µ started at x agree on FTj for each j = 1, 2,... . We prove this claim by induction. 1 By Lemma A.2 applied to Pl and P , Pl is the law of an α-stable process with drift µ|B(x,ρ) started at x. By the uniqueness of the α-stable process with drift µ|B(x,ρ) (Proposition 5.8), both 1 Pl are equal P . Hence the restriction of P1 and P2 to FT must be the same. We have shown the claim for j = 1. = Moreover, if P1 P2 on FTj , by the same argument as that in the proof of [2, Theorem VI.3.4] and the uniqueness argument in the previous paragraph, we have

 Tj+1  Tj+1 −λr = −λr EP1 e f (Xr )dr EP2 e f (Xr )dr 0 0 whenever f is bounded continuous and λ > 0. As in the proof of [2, Theorem VI.3.2], this = implies that P1 P2 on FTj . We proved the claim. Using Corollary 4.5 and the argument similar to that used in the proof of Theorem 1.4, we can show that Tj ↑ ∞. Therefore P1 = P2 on F∞.

References

[1] D. Applebaum, Levy´ Processes and Stochastic Calculus, second ed., Cambridge University Press, Cambridge, 2009. [2] R.F. Bass, Diffusions and Elliptic Operators, Springer, 1998. [3] R.F. Bass, Stochastic differential equations with jumps, Probab. Surv. 1 (2004) 1–19. [4] R.F. Bass, Z.-Q. Chen, Brownian motion with singular drift, Ann. Probab. 31 (2) (2003) 791–817. [5] P. Billingsley, Conditional distributions and tightness, Ann. Probab. 2 (1974) 480–485. [6] R.M. Blumenthal, R.K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960) 263–273. [7] R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory, Academic Press, 1968. [8] K. Bogdan, T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (2007) 179–198. [9] Z.-Q. Chen, P. Kim, R. Song, Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation, Ann. Probab. 40 (6) (2012) 2483–2538. [10] Z.-Q. Chen, L. Wang, Uniqueness of stable processes with drift, arXiv:1309.6414. [11] S.N. Ethier, T.G. Kurtz, Markov Processes, Characterization and Convergence, John Wiley & Sons, New York, 1986. [12] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. [13] B. Grigelionis, On the relative compactness of sets of probability measures in D[0, ∞), Lith. Math. J. 13 (1973) 576–586. 2516 P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516

[14] S.-W. He, J.-G. Wang, J.-A. Yan, Semimartingale Theory and Stochastic Calculus, CRC Press, Boca Raton, FL, 1992. [15] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. [16] T. Jakubowski, Fractional Laplacian with singular drift, Studia Math. 207 (2011) 257–273. [17] T. Jakubowski, K. Szczypkowski, Time-dependent gradient perturbations of fractional Laplacian, J. Evol. Equ. 10 (2010) 319–339. [18] P. Kim, R. Song, Two-sided estimates on the density of Brownian motion with singular drift, Illinois J. Math. 50 (2006) 635–688. [19] P. Kim, R. Song, Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains, Math. Ann. 339 (2007) 135–174. [20] T. Komatsu, Pseudodifferential operators and Markov processes, J. Math. Soc. Japan 36 (1984) 387–418. [21] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge, 1995. [22] R. Song, Feynman–Kac semigroups with discontinuous additive functionals, J. Theoret. Probab. 8 (1995) 727–762. [23] L. Wu, Modeling financial security returns usingevy L´ processes, in: J. Birge, V. Linetsky (Eds.), Handbooks in Operations Research and Management Science: Financial Engineering, Vol. 15, North-Holland, 2008, pp. 117–162.