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

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.