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

Abstract

Let \(\mu=(\mu^1, \dots, \mu^d)\) be such that each \(\mu^i\) is a signed measure on R d belonging to the Kato class K d, 1. A Brownian motion in R d with drift \(\mu\) is a diffusion process in R d whose generator can be informally written as \(\frac12\Delta+\mu\cdot\nabla\) . When each \(\mu^i\) is given by U i(x)dx for some function U i, a Brownian motion with drift \(\mu\) is a diffusion in R d with generator \(\frac12\Delta+U\cdot\nabla\) . 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 \(\Delta\) is replaced by a uniformly elliptic divergence form operator \(\sum_{i,j=1}^{d} \partial_i (a_{ij} \partial_j)\) with C 1 coefficients or a uniformly elliptic non-divergence form operator \(\sum_{i,j=1}^{d} a_{ij} \partial_i \partial_j\) with C 1 coefficients.