Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient

Abstract

We provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficient and a Burgers type nonlinearity. This generalizes previous results in the simpler cases of additive or affine noise. The basic tool is a discrete version of a two sided stochastic integral which allows a new formulation for the derivatives of these solutions. We show that this can be used to generalize the weak order analysis performed in [20]. The tools we develop are very general and can be used to study many other examples of applications.