Uniqueness of stable processes with drift

Abstract

Suppose that d ≥ 1 and α ∈ (1, 2). Let Y be a rotationally symmetric α-stable process on R d and b a R d -valued measurable function on R d belonging to a certain Kato class of Y . WeThroughout this paper, unless otherwise stated, d ≥ 1 and α ∈ (1, 2). A rotationally symmetric α-stable process Y in R d is a Lévy process with characteristic function given by(1.1) The infinitesimal generator of Y is the fractional Laplacian ∆ α/2 := −(−∆) α/2 . Here we use ":=" to denote a definition. Denote by B(x, r) the open ball in R d centered at x ∈ R d with radius r > 0 and dx the Lebesgue measure on R d . Definition 1.1. For a real-valued function f on R d and r > 0, define M α f (r) := sup x∈R d B(x,r) |f (y)| |x − y| d+1−α dy. (1.2) A function f on R d is said to belong to the Kato class K d,α−1 if lim r↓0 M α f (r) = 0.Using Hölder's inequality, it is easy to see that for every p > d/(α−1), L ∞ (R d ; dx)+L p (R d ; dx) ⊂ K d,α−1 . Throughout this paper we will assume b = (b 1 , · · · , b d ) is a R d -valued function on R d such that |b| ∈ K d,α−1 .

Stable process with singular drift

“…In that paper, they showed that, when µ(d x) = b(x)d x is absolutely continuous and belongs to the Kato class K d,α−1 , the martingale problem for L = ∆ α/2 + b · ∇ is well posed and the SDE d X t = d S t + b(X t )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].…”

“…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 very recently in [10] Chen and Wang studied on a related topic. In that paper, they showed that, when µ(d x) = b(x)d x is absolutely continuous and belongs to the Kato class K d,α−1 , the martingale problem for L = ∆ α/2 + b · ∇ is well posed and the SDE d X t = d S t + b(X t )dt has a unique weak solution in the "classical" sense.…”

Hölder Estimates for Nonlocal-Diffusion Equations with Drifts

“…For results on the well-posedness of the martingale problems for L b t , we refer the reader to [1,7,8] and the references therein. We remark here that if…”

Brownian Motion with Singular Time-Dependent Drift

“…Therefore, we expect to obtain a similar expression of S λ like the identity (4). However, we first have to identify the class of drifts for which the term b(t, ·) · ∇ is "small enough" compared to ∂/∂t + △/2 and the perturbation argument works.…”

“…Recently, Chen and Wang [4] studied rotationally symmetric α-stable process with a drift belonging to the Kato class K d,α−1 (see [4, Definition 1.1]); they proved the existence and uniqueness, in the weak sense, of such a process and established sharp two-sided estimates for its heat kernel. Sharp two-sided Dirichlet heat kernel estimates for such a drifted α-stable process were derived in [3].…”