Optimal convergence order for multi-scale stochastic Burgers equation

TL;DR

This paper establishes the optimal strong and weak convergence rates for multi-scale stochastic Burgers equations, overcoming nonlinear challenges with new techniques, and provides the first such results for these complex SPDEs.

Contribution

It introduces novel methods to achieve optimal convergence orders for multi-scale stochastic Burgers equations with highly nonlinear terms.

Findings

Strong convergence order is 1/2.

Weak convergence order is 1.

Developed new techniques to handle nonlinearities.

Abstract

In this paper, we study the strong and weak convergence rates for multi-scale one-dimensional stochastic Burgers equation. Based on the techniques of Galerkin approximation, Kolmogorov equation and Poisson equation, we obtain the slow component strongly and weakly converges to the solution of the corresponding averaged equation with optimal orders 1/2 and 1 respectively. The highly nonlinear term in system brings us huge difficulties, we develop new technique to overcome these difficulties. To the best of our knowledge, this work seems to be the first result in which the optimal convergence orders in strong and weak sense for multi-scale stochastic partial differential equations with highly nonlinear term.