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

In this paper, we study properties of the dual process and Schrödinger-type operators of a non-symmetric diffusion with measure-valued drift. Let μ = (μ, . . . , μ) be such that each μ is a signed measure on R belonging to the Kato class Kd,1. A diffusion with drift μ is a diffusion process in R whose generator can be informally written as L + μ · ∇ where L is a uniformly elliptic differential operator. When each μ is given by U (x)dx for some function U , a diffusion with drift μ is a diffusion in R with generator L + U · ∇. In [14, 15], we have already studied properties of diffusions with measure-value drifts in bounded domains. In this paper we discuss the potential theory of the dual process and Schrödinger-type operators of a diffusion with measure-valued drift. We show that a killed diffusion process with measure-valued drift in any bounded domain has a dual process with respect to a certain reference measure. For an arbitrary bounded domain, we show that a scale invariant Harnack inequality is true for the dual process. We also show that, if the domain is bounded C, the boundary Harnack principle for the dual process is true and the (minimal) Martin boundary for the dual process can be identified with the Euclidean The research of this author is supported in part by a joint US-Croatia grant INT 0302167.