Global regularity and bounds for solutions of parabolic equations for probability measures

Given a second-order parabolic operator Lu(t,x) := (partial derivative u(t,x))/(partial derivative t) + a(ij) (t,x)partial derivative x(i) partial derivative(xj) u(t,x) + at b(i)(t,x)partial derivative(xi)u(t,x), we consider the weak parabolic equation L*mu = 0 for Borel probability measures on (0, 1) x R-d. The equation is understood as the equality integral((0, 1)) (x Rd) Lu d mu = 0 for all smooth functions u with compact support in (0, 1) x Rd. This equation is satisfied for the transition probabilities of the diffusion process associated with L. We show that under broad assumptions, A has the form mu = rho(t, x) dt dx, where the function x -> rho(t, x) is Sobolev, vertical bar del(x)rho(x, t)vertical bar(2)/rho(t, x) is Lebesgue integrable over [0, tau] x R-d, and rho is an element of L-p ([0, tau] x R-d) for all p is an element of [1, +infinity) and tau < 1. Moreover, a sufficient condition for the uniform boundedness of rho on [0, tau] x R-d is given.