## Illinois Journal of Mathematics

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

#### 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

https://projecteuclid.org/euclid.ijm/1258059487

Digital Object Identifier
doi:10.1215/ijm/1258059487

Mathematical Reviews number (MathSciNet)
MR2247841

Zentralblatt MATH identifier
1110.60071

#### 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