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Macroscopic reduction for stochastic reaction-diffusion equations. (English) Zbl 1284.60131
Summary: The macroscopic behaviour of dissipative stochastic partial differential equations usually can be described by a finite-dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic reaction-diffusion equations by artificially separating the system into two distinct slow and fast time parts. An averaging method and a deviation estimate show that the macroscopic reduced model should be a stochastic ordinary equation that includes emergent random effects transmitted from the microscopic scales due to the nonlinear interaction. Numerical simulations of an example stochastic reaction-diffusion equation verifies the predictions of this stochastic modelling theory. This theory empowers us to better model the dynamics of complex stochastic systems on a large time scale.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K57 Reaction-diffusion equations
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