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 $$c_1{e}^{-c_{2}t}{t}^{-\frac{d}{2}}{e}^{-\frac{c_3{∣ x-y∣ }^2}{2t}}\le q^{\mu }\left(t,x,y\right)\le c_4{e}^{c_{5}t}{t}^{-\frac{d}{2}}{e}^{-\frac{c_6{∣ x-y∣ }^2}{2t}}$$ and $$∣ {\nabla }_x{q}^{\mu }\left(t,x,y\right)∣ \le c_7{e}^{c_{8}t}{t}^{-\frac{d+1}{2}}{e}^{-\frac{c_9{∣ x-y∣ }^2}{2t}}$$ 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 $$\begin{array}{c}{C}_1\left(1\wedge \frac{\rho \left(x\right)}{\sqrt{t}}\right)\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right)t^{-\frac{d}{2}}{e}^{-\frac{C_2{∣ x-y∣ }^2}{t}}\le q^{\mu ,D}\left(t,x,y\right)\\ \le C_3\left(1\wedge \frac{\rho \left(x\right)}{\sqrt{t}}\right)\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right)t^{-\frac{d}{2}}{e}^{-\frac{C_4{∣ x-y∣ }^2}{t}}\\ \end{array}$$ and $$∣ {\nabla }_x{q}^{\mu ,D}\left(t,x,y\right)∣ \le C_5\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right)t^{-\frac{d+1}{2}}{e}^{-\frac{C_4{∣ x-y∣ }^2}{t}}$$ 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 $X^D$.