Asymptotic Analysis 97 (2016) 301–327 301 DOI 10.3233/ASY-151355 IOS Press Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains Mogtaba Mohammed a,b,∗ and Mamadou Sango a a Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa E-mail: mamadou.sango@up.ac.za b Department of Mathematics, Sudan University of Science and Technology, Khartoum 11111, Sudan E-mail: mogtaba.mohammed@gmail.com Dedicated to Professors Doina Cioranescu and Patrizia Donato, Shining Queens of Homogenization Theory. Abstract. In this paper, we investigate a linear hyperbolic stochastic partial differential equation (SPDE) with rapidly oscillating ǫ -periodic coefficients in a domain with small holes (of size-ǫ ) under Neumann conditions on the boundary of the holes and Dirichlet condition on the exterior boundary. When the number of these holes approach infinity, i.e. their sizes approach zero, the homogenized problem is a hyperbolic SPDE with constant coefficients in the domain without perforations. Moreover the convergence of the associated energy to that of the homogenized system is established. Keywords: homogenization, hyperbolic SPDEs, Neumann problem, perforated domains, probabilistic compactness results 1. Introduction and setting of the problem Homogenization is a mathematical theory aimed at understanding the behavior of processes that take place in heterogeneous media with highly oscillating heterogeneities. These heterogeneous materials 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 media lead 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 solving 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. * Corresponding author. E-mail: mogtaba.mohammed@gmail.com. 0921-7134/16/$35.00 © 2016 – IOS Press and the authors. All rights reserved