Bounded Bessel processes and Ferrari-Spohn diffusions

Abstract

We introduce a new diffusion process which arises as the $n\to \mathrm{\infty }$ 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 [9]. 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.