Global Regularity and Bounds for Solutions of Parabolic Equations for Probability Measures

Abstract

Given a second order parabolic operator $$Lu(t,x):=\frac{\partial u(t,x)}{\partial t}+a^{ij}(t,x){\partial }_{x_i}{\partial }_{x_j}u(t,x)+b^i(t,x){\partial }_{x_i}u(t,x),$$ we consider the weak parabolic equation $L^{\ast }\mu =0$ for Borel probability measures on $(0,1)\times {\mathbb{R}}^d$. The equation is understood as the equality $${\int }_{(0,1)\times {\mathbb{R}}^d}Lud\mu =0$$ for all smooth functions $u$ with compact support in~$(0,1)\times {\mathbb{R}}^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with~$L$. We show that under broad assumptions $\mu$ has the form $\mu =\varrho (t,x)dtdx$, where the function $x\mapsto \varrho (t,x)$ is Sobolev, $|{\nabla }_x\varrho (x,t){|}^2/\varrho (t,x)$ is Lebesgue integrable over $[0,\tau ]\times {\mathbb{R}}^d$, and $\varrho \in L^p([0,\tau ]\times {\mathbb{R}}^d)$ for all $p\in [1,+\infty )$ and $\tau <1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,\tau ]\times {\mathbb{R}}^d$ is given.

11 pages; BiBoS-Preprint No. 05-11-198; to appear in Th. Prob. Appl