Stratonovich–Khasminskii averaging principle for multiscale random evolution equations

Multiscale phenomena are ubiquitous in nature and arise in a wide range of complex systems governed by random evolution equations. In this paper, we investigate a class of multiscale random evolution equations driven by a rapidly oscillating external force and a rapidly oscillating random noise. Our main objective is to analyze the asymptotic behavior of the system. To this end, we establish a Stratonovich–Khasminskii type averaging principle. Specifically, we show that the solutions of the multiscale random evolution equations converge in distribution (or in probability) to the solution of a stochastic evolution equation driven by a Wiener process. The results developed in this work can be applied to various types of physical models, such as the reaction–diffusion equation, the Burgers equation, the Ginzburg–Landau equation, and the two-dimensional Navier–Stokes system. This study not only extends and unifies several existing results in the literature, but also provides a general analytical framework for investigating the asymptotic behavior of a broad class of multiscale random evolution equations.