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

Positive self-similar Markov processes (pssMp) are positive Markov processes that satisfy the scaling property and it is known that can be represented as the exponential of a time-changed Lévy 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 Lévy 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 Lévy 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 Lévy process, and processes that can be represented as a function of a time-changed Lévy 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 Lévy process appears naturally and has a nice interpretation in term of the self-similar fragmentation process. To prove the general representations in dimension 1 and 2, some of our arguments work in the context of a general state space and show that, under some conditions, we can exhibit a topological group structure on the state space of a Markov process that satisfies general self-similarity properties, which allows to write a Lamperti type representation for this process in term of a Lévy process on the group