Let be such that each is a signed measure on belonging to the Kato class . The existence and uniqueness of a continuous Markov process on , called a Brownian motion with drift , was recently established by Bass and Chen. In this paper we study the potential theory of . We show that has a continuous density and that there exist positive constants , , such that and for all . We further show that, for any bounded domain , the density of , the process obtained by killing upon exiting from , has the following estimates: for any , there exist positive constants , such that and for all , where is the distance between and . Using the above estimates, we then prove the parabolic Harnack principle for and show that the boundary Harnack principle holds for the nonnegative harmonic functions of . We also identify the Martin boundary of .
"Two-sided estimates on the density of Brownian motion with singular drift." Illinois J. Math. 50 (1-4) 635 - 688, 2006. https://doi.org/10.1215/ijm/1258059487