In this paper, we consider Hessian estimates of solutions of the Cauchy problem for parabolic PDEs with the Ornstein-Uhlenbeck operator. Our upper estimate on the Hessian matrix of solutions is a generalization of the result of Kružkov . On the other hand, our lower estimate on the Hessian matrix of solutions is best possible in some sense.
"Hessian estimates for viscous Hamilton-Jacobi equations with the Ornstein-Uhlenbeck operator." Differential Integral Equations 18 (12) 1383 - 1396, 2005.