Optimal convergence rates in the averaging principle for slow-fast SPDEs driven by multiplicative noise

In this paper, we study a class of slow-fast stochastic partial differential equations with multiplicative Wiener noise. Under some appropriate conditions, we prove the slow component converges to the solution of the corresponding averaged equation with optimal orders 1/2 and 1 in the strong and weak sense respectively. The main technique is based on the Poisson equation.
Comment: 39 pages, we do some corrections and add some remarks in the previuos version