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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 probabilitymeasures on $(0,1)\times\mathbb{R}^d$. The equation is understood as theequality $$ \int_{(0,1)\times\mathbb{R}^d} Lu d\mu =0 $$ for all smooth functions $u$ with compact supportin~$(0,1)\times\mathbb{R}^d$. This equation is satisfied for the transitionprobabilities 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 forthe uniform boundedness of $\varrho$ on $[0,\tau]\times\mathbb{R}^d$ is given.

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