Title: Two-sided estimates on the density of Brownian Motion with singular drift
Speaker: Professor Renming Song
Speaker Info: University of Illinois, Urbana-Champaign
Brief Description:
Special Note:
Abstract:
Let $d\ge 3$ and let $\mu=(\mu1, \dots, \mu^d)$ with each $\mu^i$ being a signed measure on $R^d$ satisfying $$ \lim_{r\to 0}\sup_{x\in R^d}\int_{B(x, r)}\frac{|\mu^i|(dy)}{|x-y|^{d-1}} =0. $$ The existence and uniqueness of a continuous Markov process $X$ on $\R^d$, called a Brownian motion with singular drift $\mu$, was established by Bass and Chen recently. In this paper we study the potential theory of $X$. We show that $X$ has a density $q^{\mu}$ and that there exist positive constant $c_i$, $i=1, \cdots, 9$ such that $$ c_1e^{-c_2 t}t^{-\frac{d}{2}} e^{-\frac{c_3 |x-y|^2}{2t}} \le q^\mu(t, x, y)\le c_4 e^{c_5 t}t^{-\frac{d}{2}}e^{-\frac{c_6|x-y|^2}{2t}} $$ and $$ | abla_x q^\mu(t, x, y)|\le c_7 e^{c_8 t}t^{-\frac{d+1}{2}}e^{-\frac{c_9|x-y|^2}{2t}} $$ for all $(t, x, y)\in (0, \infty)\times \R^d\times \R^d$. We further show that, for every bounded $C^{1,1}$-domain $D$, the density $q^{\mu, D}$ of $X^D$, the process obtained from $X$ by killing upon exiting from $D$, has the following estimates: For any $T>0$, there exist positive constants $C_i$, $i=1, \cdots, 5 $ such that $$ C_1(1\wedge \frac{\rho(x)}{\sqrt t})(1\wedge \frac{\rho(y)}{\sqrt t}) t^{-\frac{d}2} e^{-\frac{C_2|x-y|^2}t} \le q^{\mu, D}(t, x, y)\le C_3(1\wedge \frac{\rho(x)}{\sqrt t})(1\wedge \frac{\rho(y)}{\sqrt t}) t^{-\frac{d}2}e^{-\frac{C_4|x-y|^2}{t}} $$ and $$ | abla_x q^{\mu, D}(t, x, y)|\le C_5(1\wedge \frac{\rho(y)}{\sqrt t}) t^{-\frac{d+1}2}e^{-\frac{C_4|x-y|^2}{t}} $$ where $\rho(x)$ is the distance between $x$ and $\partial D$. Using the above estimates, we prove a parabolic Harnack inequality for $X$ and boundary Harnack inequality for nonnegative harmonic functions of $X$.Date: Monday, February 7, 2005