Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator

Zili, Mounir; Zougar, Eya

doi:10.15559/19-VMSTA139

Modern Stochastics: Theory and Applications

Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator

Volume 6, Issue 3 (2019), pp. 345–375

Mounir Zili

Eya Zougar

https://doi.org/10.15559/19-VMSTA139

Pub. online: 3 October 2019 Type: Research Article

Open Access

Received
25 April 2019

Revised
3 September 2019

Accepted
3 September 2019

Published
3 October 2019

Abstract

We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein–Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.

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Keywords

Stochastic partial differential equations divergence form piecewise constant coefficients fundamental solution Stein-Malliavin calculus almost sure central limit theorem

MSC2010

60H15 60G15 60H05 35A08

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