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Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions. (English) Zbl 1126.35112

The authors consider a microscopic heterogeneous system under random influence. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. The system is modeled by a stochastic partial differential equation defined on a perforated domain with small holes (obstacles or heterogeneities), which is coupled with random dynamical boundary conditions on the boundaries of these small holes. The authors derive a homogenized macroscopic model for that microscopic heterogeneous stochastic system. The homogenized model is a new stochastic PDE defined on a unified domain without small holes, and with a static boundary condition only. The validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes approaches zero.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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