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 .