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

Abstract

For μ = ( μ 1 , … , μ d ) with each μ i being a signed measure on R d belonging to the Kato class K d , 1 , a diffusion with drift μ is a diffusion process in R d whose generator can be formally written as L + μ ⋅ ∇ where L is a uniformly elliptic differential operator. When each μ i is given by U i ( x ) d x for some function U i , a diffusion with drift μ is a diffusion in R d with generator 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 C 1 , 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 h -conditioned Brownian motion with h 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 ⋅ ∇ + ν where L is uniformly elliptic, μ is a vector-valued signed measure on R d and ν is a signed measure in R d . 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.