Optimal Convergence Rates in the Averaging Principle for Slow–Fast SPDEs Driven by Multiplicative Noise

Abstract

In this paper, the averaging principle is researched for slow–fast stochastic partial differential equations driven by multiplicative noises. The optimal orders for the slow component that converges to the solution of the corresponding averaged equation have been obtained by using the Poisson equation method under some appropriate conditions. More precisely, the optimal orders are 1/2 and 1 for the strong and weak convergences, respectively. It is worthy to point that two kinds of strong convergence are studied here and the stronger one of them answers an open question by Bréhier in [3, Remark 4.9].

Keywords

Stochastic partial differential equations / Averaging principle / Slow–fast / Poisson equation / Strong and weak convergence rates / Multiplicative noise

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Funding

National Natural Science Foundation of China(11931004)

Priority Academic Program Development of Jiangsu Higher Education Institutions

Qinglan Project of Jiangsu Province of China