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

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. Homogenization is a mathematical theory aimed at understanding the behavior of processes that take place in heterogeneous media with highly oscillating heterogeneities at the microscopic level using prop- erties of the homogeneous media obtained by homogenizing these materials. These heterogeneous ma- terials consist of finely mixed different components like soil, paper, concrete for building, fibreglass, materials used in the manufacturing of high tech equipments such as planes, rockets and so on. This signifies that almost everything around us in real life is a heterogeneous material. The physical problems described on heterogeneous materials such as heat, mechanical constraints, flow of fluids in these me- dia leads to the study of PDEs with highly oscillating coefficients depending on macroscopic scales or boundary value problems for PDEs in domain with fine grained boundaries. The main obstacle in solv- ing these problems arises either from the character of the domain or the presence of high oscillations in the coefficients of the governing equation. To this end, it is expensive to compute solutions to these type of problems. Numerical methods have proved inefficient in solving such problems due to the fact that even the most advanced parallel computers are unable to simulate schemes related to the physically interesting such problems.

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