Large Deviations and Exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts

TL;DR

This paper investigates large deviations and exit times for a class of reflected McKean-Vlasov diffusions with self-stabilizing and superlinear drift terms, establishing existence, uniqueness, and probabilistic behavior without boundedness assumptions.

Contribution

It introduces new existence and uniqueness results for reflected McKean-Vlasov equations with non-Lipschitz coefficients and proves large deviations principles and exit-time laws.

Findings

Established existence and uniqueness for the class of equations.

Proved a Freidlin-Wentzell type Large Deviations Principle.

Derived Eyring-Kramer's law for exit-times.

Abstract

We study a class of reflected McKean-Vlasov diffusions over a convex domain with self-stabilizing coefficients. This includes coefficients that do not satisfy the classical Wasserstein Lipschitz condition. Further, the process is constrained to a (not necessarily bounded) convex domain by a local time on the boundary. These equations include the subclass of reflected self-stabilizing diffusions that drift towards their mean via a convolution of the solution law with a stabilizing potential. Firstly, we establish existence and uniqueness results for this class and address the propagation of chaos. We work with a broad class of coefficients, including drift terms that are locally Lipschitz in spatial and measure variables. However, we do not rely on the boundedness of the domain or the coefficients to account for these non-linearities and instead use the self-stabilizing properties. We prove a Freidlin-Wentzell type Large Deviations Principle and an Eyring-Kramer's law for the exit-time from subdomains contained in the interior of the reflecting domain.