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

Abstract

Let μ = μ 1 ⋯ μ d be such that each μ i is a signed measure on \R d belonging to the Kato class \K d , 1 . The existence and uniqueness of a continuous Markov process X on \R d , 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 q μ and that there exist positive constants c i , i = 1 , ⋯ , 9 , such that c 1 e - c 2 t t - d 2 e - c 3 x - y 2 2 t ≤ q μ t x y ≤ c 4 e c 5 t t - d 2 e - c 6 x - y 2 2 t and ∇ x q μ t x y ≤ c 7 e c 8 t t - d + 1 2 e - c 9 x - y 2 2 t for all t x y ∈ 0 ∞ × \R d × \R d . We further show that, for any bounded C 1 , 1 domain D , the density q μ , D of X D , the process obtained by killing X upon exiting from D , has the following estimates: for any T & gt ; 0 , there exist positive constants C i , i = 1 , ⋯ , 5 , such that C 1 1 ∧ ρ ⁡ x t 1 ∧ ρ ⁡ y t t - d 2 e - C 2 x - y 2 t ≤ q μ , D t x y ≤ C 3 1 ∧ ρ ⁡ x t 1 ∧ ρ ⁡ y t t - d 2 e - C 4 x - y 2 t and ∇ x q μ , D t x y ≤ C 5 1 ∧ ρ ⁡ y t t - d + 1 2 e - C 4 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 X D .