Stable process with singular drift

Suppose that d≥2d≥2 and α∈(1,2)α∈(1,2). Let μ=(μ1,…,μd)μ=(μ1,…,μd) be such that each μiμi is a signed measure on RdRd belonging to the Kato class Kd,α−1Kd,α−1. In this paper, we consider the stochastic differential equation dXt=dSt+dAt,dXt=dSt+dAt, where StSt is a symmetric αα-stable process on RdRd and for each j=1,…,dj=1,…,d, the jjth component Atj of AtAt is a continuous additive functional of finite variation with respect to XX whose Revuz measure is μjμj. The unique solution for the above stochastic differential equation is called an αα-stable process with drift μμ. We prove the existence and uniqueness, in the weak sense, of such an αα-stable process with drift μμ and establish sharp two-sided heat kernel estimates for such a process.