Two-sided estimates on the density of Brownian motion with singular drift

Abstract

Let μ = (μ1,... ,μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,1. The existence and uniqueness of a continuous Markov process X on Rd, called a Brownian motion with drift μ, was recently established by Bass and Chen. In this paper we study the potential theory of X. We show that X has a continuous density qμ and that there exist positive constants ci, i = 1, ⋯ , 9, such that c1e-c2tt-d/2e-c3|x-y|2/2t ≤ qμ (t,x,y) ≤ c4ec5tt-d/2 e-c6|x-y|2/2t and |∇xqμ(t,x,y)| ≤ c7ec8t t-d+1/2 e-c9|x-y|2/2t for all (t,x,y) ∈ (0, ∞) × Rd × Rd. We further show that, for any bounded C1,1 domain D, the density q μ,D of XD, the process obtained by killing X upon exiting from D, has the following estimates: for any T > 0, there exist positive constants Ci, i = 1, ⋯,5, such that C1(1 Λ ρ(x)/√t)(1 Λ ρ(y)/√t)t-d/2e -C2|x-y|2/t ≤ qμ,D (t,x,y) ≤ C3 (1 Λ ρ(x)/√t)(1 Λ ρ(y)/√t)t-d/2e -C4|x-y|2/t and |∇ xqμ,D (t,x,y)| ≤ C5(1Λ ρ(y)/√t)t-d+1/2e -C4|x-y|2/t for all (t,x,y) ∈ (0, T] × D × D, where ρ(x) is the distance between x and ∂D. Using the above estimates, we then prove the parabolic Harnack principle for X and show that the boundary Harnack principle holds for the nonnegative harmonic functions of X. We also identify the Martin boundary of XD. © 2006 University of Illinois.