Two-sided Estimates on the Density of Brownian Motion with Singular Drift

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 2 e− c3|x−y| 2 2t ≤ q(t, x, y) ≤ c4e5t d 2 e− c6|x−y| 2 2t and |∇xq(t, x, y)| ≤ c7e8t 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 qμ,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 ∧ ρ(x) √ t )(1 ∧ ρ(y) √ t )t− d 2 e− C2|x−y| 2 t ≤ q(t, x, y) ≤ C3(1 ∧ ρ(x) √ t )(1 ∧ ρ(y) √ t )t− d 2 e− C4|x−y| 2 t and |∇xq(t, x, y)| ≤ C5(1 ∧ ρ(y) √ t )t− d+1 2 e− 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. Received May 9, 2005; received in final form December 5, 2005. 2000 Mathematics Subject Classification. Primary 58C60, 60J45. Secondary 35P15, 60G51, 31C25. The research of the second author is supported in part by a joint US-Croatia grant INT 0302167. c ©2006 University of Illinois 635 636 PANKI KIM AND RENMING SONG