The ODE method for some self-interacting diffusions on non-compact spaces

TL;DR

This paper investigates the long-term behavior of self-interacting diffusions on non-compact spaces, linking their occupation measures to deterministic flows, and extends previous results to more general settings with conditions for convergence.

Contribution

It extends the analysis of self-interacting diffusions to non-compact spaces and establishes a relation between occupation measures and deterministic flows, including convergence criteria.

Findings

Established a relation between asymptotic behavior of occupation measures and deterministic flows.

Extended previous results to non-compact spaces such as  and Riemannian manifolds.

Provided sufficient conditions for convergence of occupation measures.

Abstract

Self-interacting diffusions are solutions to SDEs with a drift term depending on the process and its normalized occupation measure $\mu_t$ (via an interaction potential and a confinement potential). We establish a relation between the asymptotic behavior of $\mu_t$ and the asymptotic behavior of a deterministic dynamical flow (defined on the space of the Borel probability measures). We extend previous results on $\mathbb{R}^d$ or more generally a smooth complete connected Riemannian manifold without boundary. We will also give some sufficient conditions for the convergence of $\mu_t$. Finally, we will illustrate our study with an example on $\mathbb{R}^2$.