Asymptotic limit of fully coupled multi-scale non-linear stochastic system: the non-autonomous approximation method

Abstract

In this paper, we develop a novel argument, the non-autonomous approximation method, to seek the asymptotic limits of the fully coupled multi-scale McKean-Vlasov stochastic systems with irregular coefficients, which, as summarized in [3,Section 7], remains an open problem in the field. We provide an explicit characterization for the averaged limit of the non-linear stochastic system, where both the choice of the frozen equation and the definition of the averaged coefficients are more or less unexpected since new integral terms with respect to the measure variable appear. More importantly, in contrast with the classical theory of multi-scale systems which focuses on the averaged limit of the slow process, we propose a new perspective that the asymptotic behavior of the entire system is actually governed by the limit of the fast motion. By studying the long-time estimates of the solution of the Kolmogorov equation in Wasserstein space, we identify the limiting distribution of the fast motion of the non-linear system, which, to the best of our knowledge, is new even for the classical multi-scale It\^o SDEs. Furthermore, rates of convergence are also obtained, which are rather sharp and depend only on the regularity of the coefficients with respect to the slow variable. The innovation of our argument is to transform the non-linear system into a sequence of linear but non-autonomous systems, which is rather simple insofar as it avoids to involve the mean-field type PDEs associated with non-linear stochastic system, and at the same time, it turns out to be quite effective as it enables us to show that the strong convergence in the averaging principle of the non-linear stochastic system follows directly from the weak convergence, which significantly simplified the proof.