Singular Perturbation of Kolmogorov Equation in Gauss–Sobolev Space

Abstract

This article is concerned with the Kolmogorov equation associated to a stochastic partial differential equation with an additive noise depending on a small parameter ϵ > 0. As ϵ vanishes, the parabolic equation degenerates into a first-order evolution equation. In a Gauss–Sobolev space setting, we prove that, as ϵ ↓ 0, the solution of the Cauchy problem for the Kolmogorov equation converges in L 2(μ, H) to that of the reduced evolution equation of first-order, where μ is a reference Gaussian measure on the Hilbert space H.