Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Let μ = (μ1, . . . , μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,1. A Brownian motion in Rd with drift μ is a diffusion process in Rd whose generator can be informally written as 1 2 + μ · ∇. When each μi is given by Ui (x)dx for some function Ui , a Brownian motion with drift μ is a diffusion in Rd with generator 1 2 +U ·∇. In Kim and Song (Ill J Math 50(3):635–688, 2006), some properties of Brownian motions with measure-value drifts in bounded smooth domains were discussed. In this paper we prove a scale invariant boundary Harnack principle for the positive harmonic functions of Brownian motions with measure-value drifts in bounded Lipschitz domains. We also show that the Martin boundary and the minimal Martin boundary with respect to Brownian motions with measure-valued drifts coincide with the Euclidean boundary for bounded Lipschitz domains. The results of this paper are also true for diffusions with measure-valued drifts, that is, when is replaced by a uniformly elliptic divergence form operator ∑d i, j=1 ∂i (ai j∂ j ) with C1 coefficients or a uniformly elliptic non-divergence form operator ∑d i, j=1 ai j∂i∂ j with C1 coefficients. Mathematics Subject Classification (2000) Primary 31B25 · 60J45; Secondary 60J60 · 31C35 The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. The research of P. Kim is supported by Research Settlement Fund for the new faculty of Seoul National University. P. Kim (B) Department of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea e-mail: [email protected] R. Song Department of Mathematics, University of Illinois, Urbana, IL 61801, USA e-mail: [email protected]