Connecting Orbits in Cooperative McKean-Vlasov SDEs

TL;DR

This paper extends the theory of monotone dynamical systems to cooperative McKean-Vlasov SDEs with multiplicative noise, establishing the existence of multiple invariant measures and connecting orbits, and revealing their unstable nature.

Contribution

It introduces a novel framework for analyzing stochastic order and connecting orbits in McKean-Vlasov SDEs, broadening the scope of monotone dynamical systems theory.

Findings

Existence of multiple order-related invariant measures.

Construction of monotone connecting orbits between invariant measures.

Identification of unstable invariant measures as backward limits.

Abstract

In this work we extend the framework of monotone dynamical systems to a broad and important class of stochastic equations, namely cooperative McKean-Vlasov SDEs with multiplicative noise. Under a locally dissipative assumption, our main theorem establishes the existence of multiple order-related invariant measures in the the Wasserstein space together with monotone connecting orbits (heteroclinic orbits) between them, with respect to the stochastic order. The presence of such connecting orbits also reveals the unstable nature of those invariant measures appearing as their backward limits, a dynamical feature that has remained largely unexplored in stochastic equations. The framework applies to a wide range of classical models, including granular media equations in double-well and multi-well confining potentials with quadratic interaction, perturbed double-well landscapes, and interacting multi-species population models. Our method is based on building a monotone dynamical system that preserves the stochastic order, achieved through a cone compatible with this order and an extension of the classical Dancer-Hess connecting orbit theorem.