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

We prove Lp -parabolic a-priori estimates for ət u + ∑di,j=1 cij(t)ə2x i x j u = f on Rd+1 when the coefficients cij are 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.