$L^p$-parabolic regularity and non-degenerate Ornstein-Uhlenbeck type operators

We prove Lp -parabolic a-priori estimates for solutions to purely second order parabolic PDEs on
on R^d+1 when the coefficients cij are time dependent locally bounded functions on R. We slightly
generalize the usual parabolicity assumption and show that still Lp -estimates hold
for the second spatial derivatives of u. We also investigate the dependence of the
constant appearing in such estimates from the parabolicity constant. The proof
requires the use of the stochastic integral when p is different from 2. Finally we
extend our estimates to parabolic equations involving non-degenerate Ornstein-
Uhlenbeck type operators.