Asymptotic analysis and homogenization of invariant measures

In this paper, we study limiting behavior of the invariant measures for reaction–diffusion equations in the whole space ℝd with regular and singular perturbations. In the regular case, we show the convergence of the unique stationary solution of du𝜀 = (Δu𝜀 − 𝜀u)dt + σ(x,u𝜀)dW to a stationary solution of the limiting equation du = Δudt + σ(x,u)dW. We also consider the asymptotic behavior of the stationary solution under the perturbations of spectrum. Finally, for the singular perturbation of homogenization type, we show the weak convergence of invariant measure to its homogenized limit.