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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 2 .1; 2/. Let µ D (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 D dSt C d At ; d j where St is a symmetric α-stable process on R and for each j D 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 2 .1; 2/. When α D 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 2 .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 2 .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 VD as a way of definition. For a; b 2 R, a ^ b VD minfa; bg and a _ b VD maxfa; bg. 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 VD −.−∆/α=2 which can be defined by Z A.d; α/ ∆α=2u x VD u y − u x dy . / lim . / . // dCα ; r!0 B.x;r/c jx − yj D α−1 −d=2 dCα − 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; 1/ × d × d : (1.1) jx − yjdCα R R It follows from [8, Lemma 5] that t jr p.t; x; y/j ≍ jx − yj t−.dC2)/α ^ on .0; 1/ × d × d : (1.2) x jx − yjdC2Cα R R For any λ ≥ 0, we define Z 1 Rλ.x; y/ D e−λt p.t; x; y/dt: (1.3) 0 Then, there exists a constant C1 D C1.d; α/ > 1 such that for λ ≥ 0, −1 j − jα−d ^ −2j − j−d−α ≤ λ C1 . x y (λ x y // R .x; y/ α−d −2 −d−α ≤ C1.jx − yj ^ (λ jx − yj // (1.4) P. Kim, R. Song / Stochastic Processes and their Applications 124 (2014) 2479–2516 2481 and −1 j − jα−d−1 ^ −2j − j−d−α−1 ≤ jr λ j C1 . x y (λ x y // x R .x; y/ α−d−1 −2 −d−α−1 ≤ C1.jx − yj ^ (λ jx − yj //: (1.5) (For example, see [8, Lemmas 7 and 9].) Note that, in particular, when λ D 0 we have 0 α−d−1 jrx R .x; y/j ≍ jx − yj : Thus, Z j f j.y/ Z dy ≍ j f .y/jjr R0.x; y/jdy: sup dC1−α sup x d jx − yj d x2R B.x;r/ x2R B.x;r/ d Definition 1.1. For any function f on R , we define for r > 0, Z j f j.y/ Mα.r/ D dy: f sup dC1−α d jx − yj x2R B.x;r/ d α D 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 fFk; k ≥ 1g d whose union is R so that ν1.Fk/ C ν2.Fk/ < 1 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 C − defined on each Fk and hence on R , which will be denoted as ν and ν , respectively. We use jνj D νC C 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, Z jνj.dy/ Mα.r/ D : ν sup dC1−α d jx − yj x2R 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/ D 0. We d d say that a d-dimensional vector valued signed measure µ D (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/, 1 d p d L .R I dx/ C L .R I 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 D (µ1; : : : ; µd / is a vector valued signed Radon d measure on R belonging to the Kato class Kd,α−1. d Let '.x/ be a non-negative smooth radial function in R such that suppT'U ⊂ B.0; 1/ and R nd n −n R '.x/dx D 1. Let 'n.x/ VD 2 '.2 x/. We note that suppT'nU ⊂ B.0; 2 / and 'n.x/dx D j d 1. We will fix ' and 'n throughout this paper. For signed Radon measures µ on R with 1 ≤ j ≤ d, define Z j j Un .x/ D 'n.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/ D 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 2 Kd,α−1; j D 1;:::; d, each j d Un is a bounded and smooth function on R . d d Let Ω D D.T0; 1/; R / be the family of cadl` ag` functions from T0; 1/ to R , equipped with the Skorohod topology.
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