TY - JOUR

T1 - Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

AU - Kim, Panki

AU - Song, Renming

N1 - Copyright:
Copyright 2007 Elsevier B.V., All rights reserved.

PY - 2007/9

Y1 - 2007/9

N2 - Let equation prsented be such that each μ is a signed measure on R d belonging to the Kato class K d, 1. A Brownian motion in R d with drift μ is a diffusion process in R d whose generator can be informally written. When each μ is given by U i (x)dx for some function U i , a Brownian motion with drift μ is a diffusion in R d with generator. 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 Δ is replaced by a uniformly elliptic divergence form operator equation prsented with C 1 coefficients or a uniformly elliptic non-divergence form operator equation prsented with C 1 coefficients.

AB - Let equation prsented be such that each μ is a signed measure on R d belonging to the Kato class K d, 1. A Brownian motion in R d with drift μ is a diffusion process in R d whose generator can be informally written. When each μ is given by U i (x)dx for some function U i , a Brownian motion with drift μ is a diffusion in R d with generator. 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 Δ is replaced by a uniformly elliptic divergence form operator equation prsented with C 1 coefficients or a uniformly elliptic non-divergence form operator equation prsented with C 1 coefficients.

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U2 - 10.1007/s00208-007-0110-6

DO - 10.1007/s00208-007-0110-6

M3 - Article

AN - SCOPUS:34547292589

VL - 339

SP - 135

EP - 174

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1

ER -