A first order description of a nonlinear SPDE in the spirit of rough paths

TL;DR

This paper develops a first-order approximation framework for a nonlinear SPDE driven by Gaussian noise, extending rough path ideas to analyze the regularity of solutions and their linearizations.

Contribution

It introduces a generalized Taylor expansion for nonlinear SPDEs in divergence form, connecting rough path theory with stochastic PDE analysis.

Findings

Proves a Taylor expansion for the difference between nonlinear and linearized solutions.

Shows the regularity improvement when subtracting the linearized solution.

Aligns the analysis with rough path theory for rough drivers.

Abstract

We consider a nonlinear stochastic partial differential equation (SPDE) in divergence form where the forcing term is a Gaussian noise, that is white in time and colored in space such that the gradient of the solution is H\"older-continuous, but not differentiable. Then, we prove a generalized Taylor expansion of the difference between the solution to the SPDE and the solution to its linearization around a fixed basepoint. The result is reminiscent of the theory of (controlled) rough paths and agrees with the general observation, that, in settings with a rough driver, subtracting the solution to the linearized equation yields a more regular object.