Central Limit Type Theorem and Large Deviation Principle for Multi-Scale McKean-Vlasov SDEs

TL;DR

This paper investigates the asymptotic behavior of multi-scale McKean-Vlasov stochastic systems, establishing a central limit theorem and a large deviation principle to describe their fluctuations and rare events.

Contribution

It introduces a weak convergence framework for the CLT and LDP in multi-scale McKean-Vlasov SDEs, including explicit stochastic integral terms.

Findings

Weak convergence of scaled deviations to a limiting process

Establishment of a large deviation principle for the system

Explicit stochastic integral term in the limiting process

Abstract

In this paper, we aim to study the asymptotic behavior for multi-scale McKean-Vlasov stochastic dynamical systems. Firstly, we obtain a central limit type theorem, i.e, the deviation between the slow component $X^{\varepsilon}$ and the solution $\bar{X}$ of the averaged equation converges weakly to a limiting process. More precisely, $\frac{X^{\varepsilon}-\bar{X}}{\sqrt{\varepsilon}}$ converges weakly in $C([0,T],\RR^n)$ to the solution of certain distribution dependent stochastic differential equation, which involves an extra explicit stochastic integral term. Secondly, in order to estimate the probability of deviations away from the limiting process, we further investigate the Freidlin-Wentzell's large deviation principle for multi-scale McKean-Vlasov stochastic system. The main techniques are based on the Poisson equation for central limit type theorem and the weak convergence approach for large deviation principle.