Convergence to SPDEs in Stratonovich form

TL;DR

This paper studies how solutions to certain perturbed parabolic PDEs with oscillatory random potentials converge to solutions of SPDEs in Stratonovich form as the correlation length diminishes, specifically in low-dimensional settings.

Contribution

It demonstrates the convergence of solutions to stochastic parabolic equations with multiplicative noise in Stratonovich form for small dimensions, extending the understanding of random perturbations in PDEs.

Findings

Solutions converge to SPDEs in Stratonovich form as correlation length vanishes.

The convergence holds in dimensions smaller than the order of the elliptic operator.

In higher dimensions, solutions homogenize to deterministic equations.

Abstract

We consider the perturbation of parabolic operators of the form $\partial_t+P(x,D)$ by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator $P(x,D)$, the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with a multiplicative term that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions that are larger than or equal to the order of the elliptic pseudo-differential operator $P(x,D)$. In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in the companion paper [2]. The stochastic model is therefore valid only for sufficiently small space dimensions in this class of parabolic problems.