General self-similarity properties for Markov processes and exponential functionals of L{\'e}vy processes

TL;DR

This paper explores the generalization of Lamperti's representation for self-similar Markov processes in dimensions one and two, revealing universality in dimension one and limitations in dimension two, with implications for higher dimensions.

Contribution

It extends the classical Lamperti representation to more general self-similar Markov processes, showing universality in dimension one and a new representation involving bivariate Lévy processes in dimension two.

Findings

Dimension 1 processes can be represented as functions of time-changed Lévy processes.

Dimension 2 processes require exponential functionals of bivariate Lévy processes for representation.

Classical Lamperti representation is not universal in dimension 2.

Abstract

Positive self-similar Markov processes (pssMp) are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed L\'evy process via Lamperti representation. In this work, we are interested in the following problem: what happens if we consider Markov processes in dimension $1$ or $2$ that satisfy self-similarity properties of a more general form than a scaling property ? Can they all be represented as a function of a time-changed L\'evy process ? If not, how can Lamperti representation be generalized ? We show that, not surprisingly, a Markovian process in dimension $1$ that satisfies self-similarity properties of a general form can indeed be represented as a function of a time-changed L\'evy process, which shows some kind of universality for the classical Lamperti representation in dimension $1$. However, and this is our main result, we show that a Markovian process in dimension $2$ that satisfies self-similarity properties of a general form is represented as a function of a time-changed exponential functional of a bivariate L\'evy process, and processes that can be represented as a function of a time-changed L\'evy process form a strict subclass. This shows that the classical Lamperti representation is not universal in dimension $2$. We briefly discuss the complications that occur in higher dimensions. In dimension $2$ we present an example, built from a self-similar fragmentation process, where our representation in term of an exponential functional of a bivariate L\'evy process appears naturally and has a nice interpretation in term of the self-similar fragmentation process.