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

Let d 3 and let = (1; : : : ; d) with each i being a signed measure on Rd satisfying lim r!0 sup x2Rd Z B(x;r) jij(dy) jx yjd1 = 0: The existence and uniqueness of a continuous Markov process X on Rd, called a Brownian motion with singular drift , was established by Bass and Chen recently. In this paper we study the potential theory of X. We show that X has a density q and that there exist positive constant ci, i = 1; ; 9 such that c1e c2tt d 2 e c3jxyj 2 2t q(t; x; y) c4ec5tt d2 e c6jxyj 2 2t and jrxq(t; x; y)j c7ec8tt d+1 2 e c9jxyj 2 2t for all (t; x; y) 2 (0;1) Rd Rd. We further show that, for every bounded C1;1-domain D, the density q;D of XD, the process obtained from X by killing upon exiting from D, has the following estimates: For any T> 0, there exist positive constants Ci, i = 1; ; 5 such that C1(1 ^ (x)p