Speaker：Dr. GrégoireVechambre (NYU Shanghai)

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

Time：10:00--11:00, Wednesday, 6/20/2018

Place：Weigetang 113


Abstract：

Positive self-similar Markov processes (pssMp) are Markov processes on the real
 half-line that fulfill the scaling property. A famous result by Lamperti has shown 
that such processes can be represented as the exponential of a time-changed Lévy process. 
This result is called Lamperti representation. 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 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 
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.