Non-linear Rough Heat Equations

TL;DR

This paper develops a framework for solving non-linear rough heat equations involving Laplacian operators and rough path-driven nonlinearities, extending rough path theory to handle such complex evolution equations.

Contribution

It introduces a novel extension of rough path theory to nonlinear heat equations with rough, finite-dimensional noise terms, enabling their rigorous analysis and solution.

Findings

Established existence and uniqueness of solutions.

Extended rough path theory to nonlinear PDEs.

Provided a mathematical framework for rough heat equations.

Abstract

This article is devoted to define and solve an evolution equation of the form $dy_t=\Delta y_t dt+ dX_t(y_t)$, where $\Delta$ stands for the Laplace operator on a space of the form $L^p(\mathbb{R}^n)$, and $X$ is a finite dimensional noisy nonlinearity whose typical form is given by $X_t(\varphi)=\sum_{i=1}^N x^{i}_t f_i(\varphi)$, where each $x=(x^{(1)},...,x^{(N)})$ is a $\gamma$-H\"older function generating a rough path and each $f_i$ is a smooth enough function defined on $L^p(\mathbb{R}^n)$. The generalization of the usual rough path theory allowing to cope with such kind of systems is carefully constructed.