Stable process with singular drift

Suppose that d ≥ 2 and α ∈ ( 1 , 2 ) . Let μ = ( μ 1 , … , μ d ) be such that each μ i is a signed measure on R d belonging to the Kato class K d , α − 1 . In this paper, we consider the stochastic differential equation d X t = d S t + d A t , where S t is a symmetric α -stable process on R d and for each j = 1 , … , d , the j th component A t j of A t 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.