Global regularity and estimates for solutions of parabolic equations

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 $(0,1)\times\mathbf{R}^d$. The equation is understood as the equality
$$ \int_{(0,1)\timesR^d} Lu\,d\mu=0 $$
for all smooth functions $u$ with compact support in $(0,1)\timesR^d$. 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 $[0,\tau]\times\mathbf{R}^d$, and $\varrho\in L^p([0,\tau]\timesR^d)$ for all $p\in[1,+\infty)$ and $\tau<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,\tau]\timesR^d$ is given.