Global Regularity and Bounds for Solutions of Parabolic Equations for Probability Measures
Vladimir Igorevich Bogachev, Michael Röckner, Stanislav Valer'evich Shaposhnikov · Theory of Probability and Its Applications · 2006
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 $(0,1)\times{\bf R}^d$. The equation is understood as the equality $\int_{(0,1)\times{\bf R}^d} Lu\, d\mu =0$ for all smooth functions u with compact support in $(0,1)\times{\bf R}^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with L. We show that under broad assumptions, μ has the form $\mu=\varrho(t,x)\, dt\, dx$, where the function $x\mapsto \varrho(t,x)$ is Sobolev, $| abla_x \varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over $[0,\tau]\times{\bf R}^d$, and $\varrho\in L^p([0,\tau]\times{\bf R}^d)$ for all $p\in [1,+\infty)$ and $\tau<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,\tau]\times{\bf R}^d$ is given.