Central limit type theorem and large deviation principle for multi-scale McKean-Vlasov SDEs

Themain aim of thiswork is to study the asymptotic behavior formulti-scaleMcKeanVlasov stochastic dynamical systems. Firstly, we obtain a central limit type theorem, i.e. the deviation between the slow component X-epsilon and the solution X- of the averaged equation converges weakly to a limiting process. More precisely, X epsilon-X-|root epsilon v e converges weakly in C([0, T], R-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 when the small-noise regime parameter delta -> 0 and the time scale parameter epsilon(delta) satisfies epsilon(delta)/delta -> 0. The main techniques are based on the Poisson equation for central limit type theorem and the weak convergence approach for large deviation principle.