Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient

TL;DR

This paper develops new regularity results for Kolmogorov equations linked to SPDEs with nonlinear diffusion, extending previous work to more complex noise types and enabling better weak order analysis.

Contribution

It introduces a novel discrete stochastic integral approach for derivatives, generalizing weak order analysis for SPDEs with nonlinear diffusion coefficients.

Findings

New regularity results for solutions of Kolmogorov equations with nonlinear diffusion

Generalization of weak order analysis to complex SPDEs

Development of a versatile discrete stochastic integral tool

Abstract

We provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficients 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 [16]. The tools we develop are very general and can be used to study many other examples of applications.