Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 50, Number 1-4 (2006), 635-688.
Two-sided estimates on the density of Brownian motion with singular drift
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 .
Illinois J. Math., Volume 50, Number 1-4 (2006), 635-688.
First available in Project Euclid: 12 November 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 31C45: Other generalizations (nonlinear potential theory, etc.) 35K05: Heat equation 60G51: Processes with independent increments; Lévy processes
Kim, Panki; Song, Renming. Two-sided estimates on the density of Brownian motion with singular drift. Illinois J. Math. 50 (2006), no. 1-4, 635--688. doi:10.1215/ijm/1258059487. https://projecteuclid.org/euclid.ijm/1258059487