## 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

#### Abstract

Let $\mu =\left({\mu}^{1},\cdots ,{\mu}^{d}\right)$ be such that each ${\mu}^{i}$ is a signed measure on ${\mathrm{\backslash R}}^{d}$ belonging to the Kato class ${\mathrm{\backslash K}}_{d,1}$. The existence and uniqueness of a continuous Markov process $X$ on ${\mathrm{\backslash R}}^{d}$, called a Brownian motion with drift $\mu $, 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 ${q}^{\mu}$ and that there exist positive constants ${c}_{i}$, $i=1,\cdots ,9$, such that $${c}_{1}{e}^{-{c}_{2}t}{t}^{-\frac{d}{2}}{e}^{-\frac{{c}_{3}{\u2223 x-y\u2223}^{2}}{2t}}\le {q}^{\mu}\left(t,x,y\right)\le {c}_{4}{e}^{{c}_{5}t}{t}^{-\frac{d}{2}}{e}^{-\frac{{c}_{6}{\u2223 x-y\u2223}^{2}}{2t}}\phantom{\rule{0ex}{0ex}}$$ and $$\u2223 {\nabla}_{x}{q}^{\mu}\left(t,x,y\right)\u2223 \le {c}_{7}{e}^{{c}_{8}t}{t}^{-\frac{d+1}{2}}{e}^{-\frac{{c}_{9}{\u2223 x-y\u2223}^{2}}{2t}}\phantom{\rule{0ex}{0ex}}$$ for all $\left(t,x,y\right)\in \left(0,\infty \right)\times {\mathrm{\backslash R}}^{d}\times {\mathrm{\backslash R}}^{d}$. We further show that, for any bounded ${C}^{1,1}$ domain $D$, the density ${q}^{\mu ,D}$ of ${X}^{D}$, the process obtained by killing $X$ upon exiting from $D$, has the following estimates: for any $T\&\mathrm{gt\; ;}0$, there exist positive constants ${C}_{i}$, $i=1,\cdots ,5,$ such that $$\begin{array}{c}\phantom{\rule{2em}{0ex}}{C}_{1}\left(1\wedge \frac{\rho \left(x\right)}{\sqrt{t}}\right)\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right){t}^{-\frac{d}{2}}{e}^{-\frac{{C}_{2}{\u2223 x-y\u2223}^{2}}{t}}\le {q}^{\mu ,D}\left(t,x,y\right)\\ \le {C}_{3}\left(1\wedge \frac{\rho \left(x\right)}{\sqrt{t}}\right)\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right){t}^{-\frac{d}{2}}{e}^{-\frac{{C}_{4}{\u2223 x-y\u2223}^{2}}{t}}\phantom{\rule{2em}{0ex}}\\ \end{array}$$ and $$\u2223 {\nabla}_{x}{q}^{\mu ,D}\left(t,x,y\right)\u2223 \le {C}_{5}\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right){t}^{-\frac{d+1}{2}}{e}^{-\frac{{C}_{4}{\u2223 x-y\u2223}^{2}}{t}}\phantom{\rule{0ex}{0ex}}$$ for all $\left(t,x,y\right)\in \left(0,T\right]\times D\times D$, where $\rho \left(x\right)$ is the distance between $x$ and $\partial D$. 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