On inversions and Doob $h$-transforms of linear diffusions

TL;DR

This paper explores the relationship between linear diffusions and their Doob $h$-transforms, providing a construction of the transformed process as an inversion of the original, with detailed examples including Brownian motion with drift and Bessel processes.

Contribution

It introduces a novel construction of Doob $h$-transformed diffusions as deterministic inversions of the original process, generalizing Euclidean inversions.

Findings

Constructed $X^*$ as an inversion of $X$ with a random time change.

Extended inversion concepts to generalize Euclidean inversions.

Analyzed specific cases like Brownian motion with drift and Bessel processes.

Abstract

Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic function for the infinitesimal generator of $X$ on $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details.