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

Charles-Edouard Bréhier 1 Arnaud Debussche 2
1 PSPM - Probabilités, statistique, physique mathématique
ICJ - Institut Camille Jordan [Villeurbanne]
2 MINGUS - Multi-scale numerical geometric schemes
IRMAR - Institut de Recherche Mathématique de Rennes, ENS Rennes - École normale supérieure - Rennes, Inria Rennes – Bretagne Atlantique
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.
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Charles-Edouard Bréhier, Arnaud Debussche. Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient. Journal de Mathématiques Pures et Appliquées, Elsevier, 2018, 116, pp.193-254. ⟨10.1016/j.matpur.2018.08.010⟩. ⟨hal-01481966v2⟩

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