A first order description of a nonlinear SPDE in the spirit of rough paths
Analysis of PDEs
2022-02-03 v1 Probability
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.
Cite
@article{arxiv.2202.01127,
title = {A first order description of a nonlinear SPDE in the spirit of rough paths},
author = {Florian Kunick},
journal= {arXiv preprint arXiv:2202.01127},
year = {2022}
}