Non uniqueness of stationary measures for self-stabilizing processes

TL;DR

This paper studies self-stabilizing diffusions, revealing that their nonlinear nature can lead to multiple invariant measures due to non-convex environments, with implications for understanding complex stochastic systems.

Contribution

It demonstrates the non-uniqueness of stationary measures in self-stabilizing processes caused by non-convex landscapes and introduces generalized Laplace's method for analysis.

Findings

Multiple invariant measures can exist for self-stabilizing diffusions.

Non-convex environments are key to the non-uniqueness phenomenon.

Generalized Laplace's method aids in analyzing these measures.

Abstract

We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits to point out singular phenomenons like non uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non convex environment and requires generalized Laplace's method approximations.