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

Abstract

In this paper, we study properties of the dual process and Schrodinger-type operators of a non-symmetric diffusion with measure-valued drift. Let mu=(mu^1,..., mu^d) be such that each mu^i is a signed measure on R^d belonging to the Kato class K_{d, 1}. 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^{1,1}, 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 boundary. It is also shown that the harmonic measure for the dual process is locally comparable to that of the h-conditioned Brownian motion with h being the ground state. Under the gaugeability assumption, if the domain is bounded Lipschitz, the (minimal) Martin boundary for the Schrodinger operator obtained from the diffusion with measure-value drift can be identified with the Euclidean boundary.