Macroscopic reduction for stochastic reaction-diffusion equations. (English) Zbl 1284.60131

Wang, W.; Roberts, A. J.

IMA J. Appl. Math. 78, No. 6, 1237-1264 (2013).

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

Keywords:

stochastic reaction-diffusion equations; averaging; tightness; martingale