Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method

Abstract

In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the multiple expansion method to drive the homogenized problem. Next we use the two scale convergence method and Prokhorov's and Skorokhod's probabilistic compactness results. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove a corrector result.

Sigma-convergence of semilinear stochastic wave equations

“…The homogenization of hyperbolic stochastic partial differential equations (SPDEs) is at its infancy as evidenced by the very few number of published papers in that direction; see e.g. [31,32,33]. In the three references above the authors deal with linear hyperbolic SPDE associated to the operator…”

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

“…The homogenization of hyperbolic SPDEs was initiated in [27], [28,29], [30] where the authors studied the homogenization of Dirichlet problems for linear hyperbolic equation with rapidly oscillating coefficients using the method of the two-scale convergence pioneered by Nguetseng in [32] and developed by Allaire in [4] and [5]; they also dealt with the linear Neumann problem by means of Tartar's method and obtained the corresponding corrector results within these settings; [30] deals with a semilinear hyperbolic SPDE by Tartar's method.…”

“…We are concerned with the homogenization of the initial boundary value problem with oscillating data, referred to throughout the paper as problem (P ): du t −div (A (x) ∇u ) dt + B(t, u t )dt = f (t, x, x/ε, ∇u )dt + g(t, x, x/ε, u t )dW in (0, T ) × Q u = 0 on (0, T ) × ∂Q, u (0, x) = a (x), u t (0, x) = b (x) in Q, where u t denotes the partial derivative ∂u /∂t of u with respect to t, > 0 is a sufficiently small parameter which ultimately tends to zero, T > 0, Q is an open and bounded (at least Lipschitz) subset of R n , W = (W (t)) (t ∈ [0, T ]) an m-dimensional standard Wiener process defined on a given filtered complete probability space (Ω, F, P, (F t ) 0≤t≤T ); E denotes the corresponding mathematical expectation. For a physical motivation, we refer to [27,28], where the authors discussed real life processes of vibrational nature subjected to random fluctuations; for instance the nonlinear term B(t, u t ) stands for damping effects, the term f (t, x, x/ε, ∇u ) is the oscillating regular part of the force acting on the system and depending linearly on ∇u , while the term g(t, x, x/ε, u t )dW represents the oscillating random component of the force; it depends on u ε t . More precise assumptions on the data will be provided shortly.…”

“…Few words about the difference between the current work and previous works by the authors on homogenization of SPDEs. Compared to [27,28,29,30], the structure of problem (P ε ) is dominated by nonlinear terms such as the damping B(t, u t ), leading to L p (Q)-like norms whose combination with the predominently L 2 -like norms coming from the other terms requires special care, both in the derivation of the a priori estimates, as well as in the passage to the limit. Though, two-scale convergence method is also used in the paper [27], the model there is essentially linear.…”

“…Compared to [27,28,29,30], the structure of problem (P ε ) is dominated by nonlinear terms such as the damping B(t, u t ), leading to L p (Q)-like norms whose combination with the predominently L 2 -like norms coming from the other terms requires special care, both in the derivation of the a priori estimates, as well as in the passage to the limit. Though, two-scale convergence method is also used in the paper [27], the model there is essentially linear. The works [43,44] deal with stochastic parabolic equations in domains with fine grained boundaries, where no conditions of periodicity hold and the methodology implemented there is a stochastic counterpart of Kruslov-Marchenko's [23] and Skrypnik's [46] homogenization theories based on potiential theory; for instance the homogenized problems in [43,44] involve an additional term of capacitary type.…”