# Space and time inversions of stochastic processes and Kelvin transform

## TL;DR

This paper explores the relationship between space inversion properties of Markov processes and Kelvin transforms, providing new classes of processes with explicit inversions and analyzing their harmonic functions.

## Contribution

It establishes that space inversion properties imply Kelvin transforms for harmonic functions and identifies new classes of processes with these properties, including explicit examples.

## Key findings

- Space inversion implies Kelvin transform for harmonic functions.
- New classes of processes with space inversion properties are identified.
- Explicit inversions and excessive functions are provided for several stochastic processes.

## Abstract

Let $X$ be a standard Markov process. We prove that a space inversion property of $X$ implies the existence of a Kelvin transform of $X$-harmonic, excessive and operator-harmonic functions and that the inversion property is inherited by Doob $h$-transforms. We determine new classes of processes having space inversion properties amongst transient processes {satisfying the} time inversion property. {For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly.} We treat in details the examples of free scaled power Bessel processes, non-colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.