We consider a class of stochastic reaction-diffusion equations also having a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction, it is possible to replace the SPDE by a suitable one-dimensional stochastic differential equation. This replacement is possible under the assumption of spectral gap for the diffusion and is a result of averaging in the fast spatial transport. We also study the fluctuations around the averaged motion.
"Fast transport asymptotics for stochastic RDEs with boundary noise." Ann. Probab. 39 (1) 369 - 405, January 2011. https://doi.org/10.1214/10-AOP552