Averaging principle for semilinear slow-fast rough partial differential equations

TL;DR

This paper establishes an averaging principle for semilinear slow-fast rough partial differential equations driven by rough paths and Brownian noise, showing strong convergence of the slow component to an averaged equation.

Contribution

It extends averaging principles to semilinear PDEs with rough and stochastic noise, employing controlled rough path theory and Khasminskii's discretization.

Findings

Strong convergence of the slow component to the averaged solution.

Application of controlled rough path theory to PDEs with rough noise.

Extension of classical averaging to rough partial differential equations.

Abstract

In this paper, we investigate the averaging principle for a class of semilinear slow-fast partial differential equations driven by finite-dimensional rough multiplicative noise. Specifically, the slow component is driven by a general random $\gamma$-H\"{o}lder rough path for some $\gamma \in (1/3,1/2)$, while the fast component is driven by a Brownian rough path. Using controlled rough path theory and the classical Khasminskii's time discretization scheme, we demonstrate that the slow component converges strongly to the solution of the corresponding averaged equation under the H\"{o}lder topology.