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

TL;DR

This paper investigates entrance measures for time-inhomogeneous McKean-Vlasov SDEs, establishing existence results in broad settings and exploring periodic/quasi-periodic measures via a novel double-lift dynamical system approach.

Contribution

It introduces a general framework for existence of entrance measures and develops a double-lift method to analyze periodic and quasi-periodic measures in McKean-Vlasov SDEs.

Findings

Existence of entrance measures in broad, possibly degenerate systems.

Construction of periodic and quasi-periodic measures under time-periodic parameters.

Development of a double-lift dynamical system approach for measure analysis.

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.