Stable process with singular drift

Abstract

Suppose that d≥2 and α∈(1,2). Let μ=(μ1,…,μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,α−1. In this paper, we consider the stochastic differential equation dXt=dSt+dAt, where St is a symmetric α-stable process on Rd and for each j=1,…,d, the jth component Atj of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is μj. 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.