Diffusion hitting times and the Bell-shape

TL;DR

This paper characterizes when diffusion hitting times have bell-shaped densities, linking this property to the measure's points of increase, and relates it to Yamazato's factorization.

Contribution

It provides a necessary and sufficient condition for diffusion hitting times to be bell-shaped, connecting measure properties with density shape.

Findings

Hitting densities are bell-shaped iff the measure has infinitely many points of increase.

The result is a visual corollary to Yamazato's factorization.

Hitting time densities are unimodal and bell-shaped under specific measure conditions.

Abstract

Consider a generalized diffusion on R with speed measure m, in the natural scale. It is known that the conditional hitting times have a unimodal density function. We show that these hitting densities are bell-shaped if and only if m has infinitely many points of increase between the starting point and the hit point. This result can be viewed as a visual corollary to Yamazato's general factorization for diffusion hitting times.