Rough Stochastic PDEs

TL;DR

This paper applies rough path theory to define and analyze solutions for a class of highly irregular nonlinear stochastic PDEs of Burgers type, establishing well-posedness and stability.

Contribution

It introduces a pathwise solution framework for ill-posed SPDEs using rough paths, ensuring stability and well-posedness under various perturbations.

Findings

Pathwise solutions are well-defined for rough Burgers-type SPDEs.

Solutions are stable under noise mollification and hyperviscosity.

Generated Markov semigroups are reversible with explicit diffusion measures.

Abstract

In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill-posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough paths theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well-posed problem, which is stable under a large class of perturbations, including the approximation of the rough driving noise by a mollified version and the addition of hyperviscosity. We also show that under certain structural assumptions on the coefficients, the SPDEs under consideration generate a reversible Markov semigroup with respect to a diffusion measure that can be given explicitly.