Macroscopic reduction for stochastic reaction-diffusion equations

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.