Existence, uniqueness and exponential ergodicity under Lyapunov conditions for McKean-Vlasov SDEs with Markovian switching

TL;DR

This paper investigates the existence, uniqueness, and exponential ergodicity of solutions to McKean-Vlasov SDEs with Markovian switching, using Lyapunov conditions and truncation techniques.

Contribution

It establishes global existence and uniqueness under Lyapunov conditions for locally Lipschitz coefficients and proves exponential convergence to invariant measures.

Findings

Existence and uniqueness of solutions under Lyapunov conditions.

Exponential convergence to invariant measures in Wasserstein and total variation distances.

Applications demonstrating the theoretical results.

Abstract

The paper is dedicated to studying the problem of existence and uniqueness of solutions as well as existence of and exponential convergence to invariant measures for McKean-Vlasov stochastic differential equations with Markovian switching. Since the coefficients are only locally Lipschitz, we need to truncate them both in space and distribution variables simultaneously to get the global existence of solutions under the Lyapunov condition. Furthermore, if the Lyapunov condition is strengthened, we establish the exponential convergence of solutions' distributions to the unique invariant measure in Wasserstein quasi-distance and total variation distance, respectively. Finally, we give two applications to illustrate our theoretical results.