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.