On the local times of noise reinforced Bessel processes

TL;DR

This paper studies how noise reinforcement influences Bessel processes of certain dimensions, focusing on the asymptotic behavior of additive functionals and introducing a local time process with explicit properties.

Contribution

It introduces a local time process for noise reinforced Bessel processes and characterizes its inverse as a self-similar Markov process, providing explicit results.

Findings

Identification of the inverse local time as a self-similar Markov process

Explicit characterization of the local time process

Insights into the asymptotic behavior of additive functionals

Abstract

We investigate the effects of noise reinforcement on a Bessel process of dimension $d\in(0,2)$, and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.