The random periodic solutions for McKean-Vlasov stochastic differential equations

TL;DR

This paper investigates the existence and properties of random periodic solutions for McKean-Vlasov stochastic differential equations driven by Brownian motion, establishing well-posedness, propagation of chaos, and behavior under different dissipativity conditions.

Contribution

It introduces a comprehensive analysis of random periodic solutions for McKean-Vlasov SDEs, including proofs under both fully and partially dissipative settings, and connects particle systems with the limiting equations.

Findings

Propagation of chaos established for particle systems

Random periodic solutions exist under full dissipativity

Weak sense periodicity in partial dissipativity case

Abstract

In this paper, we study well-posedness of random periodic solutions of stochastic differential equations (SDEs) of McKean-Vlasov type driven by a two-sided Brownian motion, where the random periodic behaviour is characterised by the equations' long-time behaviour. Given the well-known connection between McKean-Vlasov SDEs and interacting particle systems, we show propagation of chaos and that the key properties of the interacting particle systems recover those of the McKean-Vlasov SDEs in the particle limit. All results in the present work are shown under two settings: fully and partially dissipative case. Each setting has its challenges and limitations. For instance, weakening full dissipativity to partial dissipativity demands stronger structural assumptions on the equations' dynamics and yields random periodic behaviour in the weak sense instead of pathwise sense (as in the full dissipativity case). The proof mechanisms are close but fundamentally different.