Bounded Bessel Processes and Ferrari-Spohn Diffusions

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 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.