Pierre Patie - Discrete self-similarity and some ergodic Markov chains
|Dates:||4 May 2022|
|Times:||15:00 - 16:00|
|What is it:||Seminar|
|Organiser:||Department of Mathematics|
Pierre Patie (Cornell University) will speak in the Probability seminar.
In this talk, resorting to the theory of group representation, we start by introducing an operator that defines the concept of discrete self-similarity, that is a scaling on a lattice. We proceed by identifying, by means of their generator, the class of Markov chains which are semi-invariant by this group action, and call them discrete self-similar Markov chains. We show that they belong to the domain of attraction of a subclass of the Lamperti (continuous) self-similar Markov processes on the positive real line. More interestingly, it turns out that the corresponding semigroups of the continuous and discrete versions satisfy a so-called gateway relation, revealing that scaling limit may be substitute to a (pseudo-) isomorphism between these operators. We explain why this unexpected fact is, in fact, natural. This gateway relation is preserved when one considers the ergodic companions of these non-reversible Markov processes. We combined this fact with some additional classification schemes that we have proposed recently to obtain several substantial analytical and ergodic properties of the discrete version such as the spectral properties, entropy decay, hypercoercivity and hypercontractivity. We emphasize that all these properties for the discrete version are transferred from the continuous analogue (the scaling limit), which is somehow unusual.
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