Macroscopic reduction for stochastic reaction-diffusion equations
W. Wang, Anthony John Roberts · IMA Journal of Applied Mathematics · 2012
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 non-linear 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.