Illinois Journal of Mathematics

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

Panki Kim and Renming Song

Full-text: Open access

Abstract

Let μ = μ 1 μ d be such that each μ i 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 c 1 e - c 2 t t - d 2 e - c 3 x - y 2 2 t q μ t x y c 4 e c 5 t t - d 2 e - c 6 x - y 2 2 t and x q μ t x y c 7 e c 8 t t - d + 1 2 e - c 9 x - y 2 2 t 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 & gt ; 0 , there exist positive constants C i , i = 1 , , 5 , such that C 1 1 ρ x t 1 ρ y t t - d 2 e - C 2 x - y 2 t q μ , D t x y C 3 1 ρ x t 1 ρ y t t - d 2 e - C 4 x - y 2 t and x q μ , D t x y C 5 1 ρ y t t - d + 1 2 e - C 4 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 .

Article information

Source
Illinois J. Math., Volume 50, Number 1-4 (2006), 635-688.

Dates
First available in Project Euclid: 12 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258059487

Digital Object Identifier
doi:10.1215/ijm/1258059487

Mathematical Reviews number (MathSciNet)
MR2247841

Zentralblatt MATH identifier
1110.60071

Subjects
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

Citation

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


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