The order of convergence in the averaging principle for slow-fast systems of stochastic evolution equations in Hilbert spaces

TL;DR

This paper establishes that the slow component in certain stochastic evolution equations converges strongly to the averaged system with an optimal order of 1/2, even with general Wiener process noise.

Contribution

It proves the strong convergence order of 1/2 for the averaging principle in slow-fast stochastic evolution equations with non-trace class noise.

Findings

Convergence order of 1/2 is optimal.

Applicable to systems with space-time white noise.

Extends results to reaction diffusion systems.

Abstract

In this work we are concerned with the study of the strong order of convergence in the averaging principle for slow-fast systems of stochastic evolution equations in Hilbert spaces with additive noise. In particular the stochastic perturbations are general Wiener processes, i.e their covariance operators are allowed to be not trace class. We prove that the slow component converges strongly to the averaged one with order of convergence $1/2$ which is known to be optimal. Moreover we apply this result to a slow-fast stochastic reaction diffusion system where the stochastic perturbation is given by a white noise both in time and space.