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

Charles-Edouard Bréhier (1, 2) , Arnaud Debussche (3, 4)

1 ICJ - Institut Camille Jordan
2 PSPM - Probabilités, statistique, physique mathématique
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 IPSO - Invariant Preserving SOlvers

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.

Keywords: Stochastic partial differential equations; Kolmogorov equations in infinite dimensions; two-sided stochastic integrals; Malliavin calculus; weak convergence rates

Domains: Probability [math.PR]; Analysis of PDEs [math.AP]

Charles-Edouard Bréhier, Arnaud Debussche. Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient. 2017. ⟨hal-01481966v1⟩