Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

TL;DR

This paper introduces a novel sticky coupling method to analyze the long-term behavior of nonlinear SDEs and McKean-Vlasov equations without confinement, revealing phase transitions and exponential convergence.

Contribution

It develops a new approach using sticky couplings to study convergence and propagation of chaos in nonlinear SDEs without confinement potentials.

Findings

Exponential convergence to equilibrium in certain regimes.

Phase transition in the dominating nonlinear SDE.

Applicability to non-gradient interaction terms.

Abstract

We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.