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

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 $L^{*}\mu=0$ for Borel probability measures on $(0,1)\times\mathbb{R}^d$. The equation is understood as the equality $$\int_{(0,1)\times\mathbb{R}^d} Lu d\mu =0$$ 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 $\mu=\varrho(t,x) dt dx$, 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.