Bounded Bessel Processes and Ferrari-Spohn Diffusions

TL;DR

This paper introduces a new diffusion process as the limit of a conditioned Bessel process, connecting it to the Ferrari-Spohn diffusion and highlighting its role as a hard edge counterpart in stochastic processes.

Contribution

It presents a novel diffusion process derived from Bessel processes conditioned to stay bounded, establishing its relation to Sturm-Liouville problems and the Ferrari-Spohn diffusion.

Findings

New diffusion process as $n\to\infty$ limit of conditioned Bessel process

Connection established between the new process and the Ferrari-Spohn diffusion

Generator relates to Sturm-Liouville problem for Bessel operator

Abstract

We introduce a new diffusion process which arises as the $n\to\infty$ limit of a Bessel process of dimension $d \ge 2$ conditioned upon remaining bounded below one until time $n$. In addition to being interesting in its own right, we argue that the resulting diffusion process is a natural hard edge counterpart to the Ferrari-Spohn diffusion of arXiv:math/0308242. In particular, we show that the generator of our new diffusion has the same relation to the Sturm-Liouville problem for the Bessel operator that the Ferrari-Spohn diffusion does to the corresponding problem for the Airy operator.