Strongly self-interacting processes on the circle

TL;DR

This paper studies the long-term behavior of self-interacting processes on the circle, including diffusion and velocity jump processes, revealing conditions for convergence or recurrence based on potential functions.

Contribution

It introduces a new analysis of self-interacting velocity jump processes and extends understanding of their long-term dynamics on the circle.

Findings

Diffusion case analyzed for specific potential functions.

Velocity jump process studied as a new class of piecewise deterministic processes.

Results show convergence or recurrence depending on potential shape.

Abstract

The purpose of this paper is to investigate the long time behaviour for a self-interacting diffusion and a self-interacting velocity jump process. While the diffusion case has already been studied for some particular potential function, the second one, which belongs to the family of piecewise deterministic processes, is new. Depending on the underlying potential function's shape, we prove either the almost sure convergence or the recurrence for a natural extended process given by a change a variable.