General self-similarity properties for Markov processes and exponential functionals of Lévy processes - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2018

General self-similarity properties for Markov processes and exponential functionals of Lévy processes

Résumé

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 Lévy 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 Lévy process, and we can see that processes which can be represented as functions of time-changed Lévy 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 Lévy process on this group.
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Dates et versions

hal-01829804 , version 1 (04-07-2018)
hal-01829804 , version 2 (11-11-2019)

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Grégoire Véchambre. General self-similarity properties for Markov processes and exponential functionals of Lévy processes. 2018. ⟨hal-01829804v1⟩
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