Rough path stability of (semi-)linear SPDEs

TL;DR

This paper establishes the stability of (semi-)linear SPDEs driven by rough path noise, including Brownian motion with Lévý area, ensuring well-defined solutions and connecting rough path solutions with classical variational solutions.

Contribution

It introduces a framework for defining and analyzing linear and semi-linear parabolic SPDEs with rough path noise, proving stability and equivalence with classical solutions in special cases.

Findings

Proved stability of (semi-)linear SPDEs in rough path metrics.

Established equivalence of rough path solutions with classical variational solutions for Brownian noise.

Extended the theory to degenerate parabolic equations with affine linear rough path noise.

Abstract

We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian motion with its L\'evy area) as rough path noise the solution coincides with the standard variational solution of the SPDE.