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

Let μ = (μ¹,... ,μ^d) be such that each μⁱ 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 c₁e^-c2tt^-d/2e^-c3|x-y|2/2t ≤ q^μ (t,x,y) ≤ c₄e^c5tt^-d/2 e^-c6|x-y|2/2t and |∇_xq^μ(t,x,y)| ≤ c₇e^c8t t^-d+1/2 e^-c9|x-y|2/2t 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 > 0, there exist positive constants C_i, i = 1, ⋯,5, such that C₁(1 Λ ρ(x)/√t)(1 Λ ρ(y)/√t)t^-d/2e ^-C2|x-y|2/t ≤ q^μ,D (t,x,y) ≤ C₃ (1 Λ ρ(x)/√t)(1 Λ ρ(y)/√t)t^-d/2e ^-C4|x-y|2/t and |∇ _xq^μ,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 X^D.