Diffusions from Infinity

TL;DR

This paper investigates diffusions on the half line with bounded expected hitting times at the origin, analyzing their behavior from infinity, fluctuations, tail distributions, and spectral properties of the killed process.

Contribution

It establishes the existence of a diffusion from infinity, characterizes its small-time behavior, and derives spectral and distributional properties of the process and its hitting times.

Findings

Process coming down from infinity is governed by a deterministic function.

Fluctuations of hitting times are asymptotically Gaussian.

Distribution of the process killed at the origin is absolutely continuous with an explicit density.

Abstract

In this paper we consider diffusions on the half line (0, $\infty$) such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity, which takes finite values at positive times. We study the behaviour of hitting times of large barriers and in a dual way, the behaviour of the process starting at infinity for small time. In particular we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion.