Entrance measures and dynamics for time-inhomogeneous McKean-Vlasov stochastic differential equations

Abstract

In this paper, we study the entrance measures of time-inhomogeneous McKean-Vlasov SDEs. The existence is obtained in great generality, where the system can be expanding globally and/or degenerate for numerous number of time intervals. When the parameters are periodic/quasi-periodic in time, we obtain the existence of periodic/asymptotic quasi-periodic measures. In this case, a double-lift of the random dynamical system first to a dynamical system on cylinder and then on the graph of reparameterized process living on the cylinder is introduced. The double-lifted system gives to a continuous dynamical system over probability measures on the cylinder, and the lifted multi-parameter measure of the asymptotic quasi-periodic measure can then lead to an invariant measure of the lifted semigroup.