On dual processes of non-symmetric diffusions with measure-valued drifts

For μ=(μ1,…,μd)μ=(μ1,…,μd) with each μiμi being a signed measure on Rd belonging to the Kato class Kd,1, a diffusion with drift μμ is a diffusion process in Rd whose generator can be formally written as L+μ⋅∇L+μ⋅∇ where LL is a uniformly elliptic differential operator. When each μiμi is given by Ui(x)dx for some function UiUi, a diffusion with drift μμ is a diffusion in Rd with generator L+U⋅∇L+U⋅∇. In [P. Kim, R. Song, Two-sided estimates on the density of Brownian motion with singular drift, Illinois J. Math. 50 (2006) 635–688; P. Kim, R. Song, Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains, Math. Ann., 339 (1) (2007) 135–174], we have already studied properties of diffusions with measure-valued drifts in bounded domains. In this paper we first show that the killed diffusion process with measure-valued drift in any bounded domain has a dual process with respect to a certain reference measure. We then discuss the potential theory of the dual process and Schrödinger-type operators of a diffusion with measure-valued drift. More precisely, we prove that (1) for any bounded domain, a scale invariant Harnack inequality is true for the dual process; (2) if the domain is bounded C1,1C1,1, the boundary Harnack principle for the dual process is valid and the (minimal) Martin boundary for the dual process can be identified with the Euclidean boundary; and (3) the harmonic measure for the dual process is locally comparable to that of the hh-conditioned Brownian motion with hh being an eigenfunction corresponding to the largest Dirichlet eigenvalue in the domain.The Schrödinger operator that we consider can be formally written as L+μ⋅∇+νL+μ⋅∇+ν where LL is uniformly elliptic, μμ is a vector-valued signed measure on Rd and νν is a signed measure in Rd. We show that, for a bounded Lipschitz domain and under the gaugeability assumption, the (minimal) Martin boundary for the Schrödinger operator obtained from the diffusion with measure-valued drift can be identified with the Euclidean boundary.