Stable process with singular drift

Suppose that d ≥ 2 and α ∈ (1, 2). Let μ = (μ, . . . , μ) be such that each μ is a signed measure on R 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 R and for each j = 1, . . . , d, the j-th component of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is μ . 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 process. AMS 2000 Mathematics Subject Classification: Primary 60J35, 47G20, 60J75; Secondary 47D07