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

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.Keywords Stochastic partial differential equations, divergence form, piecewise constant coefficients, fundamental solution, Stein-Malliavin calculus, almost sure central limit theorem 2010 MSC 60H15, 60G15, 60H05, 35A08

A note on parameter estimation for discretely sampled SPDEs

“…[CH19, BT19b, BT19a, Cho19a, CDVK19, Cho19b, KU19, KT19] and references therein. We follow the p-variation approach of [CH19], which exploit the optimal regularity of the solution combined with the power variation calculus to build consistent and asymptotically normal estimators of the unknown parameters; see also [PT07,ZZ19,KT19].…”

On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity

“…In Zili and Zougar (2018), the authors proved the existence of the mild solution to (1), they presented explicit expressions of its covariance and variance functions, and they analyzed some regularity properties of its sample paths. Then, Zili and Zougar (2019a) presented an estimation method of the parameters a 1 and a 2 appearing in (3) and, after that, in Zili and Zougar (2019b), they made a deep study of the spatial quadratic variations of the mild solution process. We make here an interesting new step in the study of SPDE (1), by introducing a notion of weak solution of this equation and then, by showing its equivalence with the already known notion of mild solution.…”