Inversion, duality and Doob $h$-transforms for self-similar Markov processes

TL;DR

This paper extends the Lamperti transformation to represent self-similar Markov processes as path transformations of Markov additive processes, establishing duality relations and inversion properties that help analyze their behavior.

Contribution

It introduces an extended Lamperti transformation for self-similar Markov processes and characterizes duality and inversion properties in this framework.

Findings

Self-similar Markov processes can be represented via Markov additive processes.

Duality between processes is characterized by reversibility of the underlying MAP.

Inversion of the process relates to the dual process, aiding in analysis of stable Lévy processes.

Abstract

We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov process $X$, with index $\alpha>0$ can be represented as a path transformation of some Markov additive process (MAP) $(\theta,\xi)$ in $S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti transformation. Let us denote by $\widehat{X}$ the self-similar Markov process which is obtained from the MAP $(\theta,-\xi)$ through this extended Lamperti transformation. Then we prove that $\widehat{X}$ is in weak duality with $X$, with respect to the measure $\pi(x/\|x\|)\|x\|^{\alpha-d}dx$, if and only if $(\theta,\xi)$ is reversible with respect to the measure $\pi(ds)dx$, where $\pi(ds)$ is some $\sigma$-finite measure on $S_{d-1}$ and $dx$ is the Lebesgue measure on $\mathbb{R}$. Besides, the dual process $\widehat{X}$ has the same law as the inversion $(X_{\gamma_t}/\|X_{\gamma_t}\|^2,t\ge0)$ of $X$, where $\gamma_t$ is the inverse of $t\mapsto\int_0^t\|X\|_s^{-2\alpha}\,ds$. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable L\'evy processes.