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

Grégoire Véchambre

Published 2018-07-05Version 1

In this work, we are interested in Markov processes that satisfy self-similarity properties of a very general form (we call them general self-similar Markov processes, or gssMp's for short) and we prove a generalized Lamperti representation for these processes. More precisely we show that, in dimension 1, a gssMp can be represented as a function of a time-changed L{\'e}vy process, which shows some kind of universality for the classical Lamperti representation in dimension 1. In dimension 2, we show that a gssMp can be represented in term of the exponential functional of a bivariate L{\'e}vy process, and we can see that processes which can be represented as functions of time-changed L{\'e}vy processes form a strict subclass of gssMp's in dimension 2. In other words, we show that the classical Lamperti representation is not universal in dimension 2. We also study the case of more general state spaces and show that, under some conditions, we can exhibit a topological group structure on the state space of a gssMp which allows to write a Lamperti type representation for the gssMp in term of a L{\'e}vy process on this group.