Non-linear Rough Heat Equations

Abstract

This article is devoted to define and solve an evolution equation of the form $d{y}_t=\Delta y_{t}dt+d{X}_t(y_t)$, 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 $X_t(\phi )={\sum }_{i=1}^N{x}_t^i{f}_i(\phi )$, where each $x=(x^{(1)},...,x^{(N)})$ is a $\gamma$-Hölder 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.

36 pages

References in corpus (1)

• Rough solutions for the periodic Korteweg-de Vries equation