Reductions and Deviations for Stochastic Partial Differential Equations under Fast Dynamical Boundary Conditions

As a model for multiscale systems under random influences on physical boundary, a stochastic partial differential equation under a fast random dynamical boundary condition is investigated. An effective equation is derived and justified by reducing the random dynamical boundary condition to a random static boundary condition. The effective system is still a stochastic partial differential equation, but is more tractable as it is only subject to the usual static, instead of dynamical, boundary condition. Furthermore, the quantitative comparison between the solution of the original stochastic system and the effective solution is provided by proving normal deviations and large deviations principles. Namely, the normal deviations are shown to be asymptotically Gaussian, while the rate and speed of the large deviations are also determined.