Dynamical Gibbs variational principles and applications

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Meeting ID: 393 527 089 878
Passcode: dsm7py

Abstract

Interacting particle systems are countable systems of locally interacting Markov processes and are often used as toy models for stochastic phenomena with an underlying spatial structure. Even though the definition of an interacting particle system often looks very simple and the major technical issues of existence and uniqueness have long been settled, it is in general surprisingly difficult to say anything non-trivial about their behavior. We study the long-time behaviour of such systems on the $d$-dimensional hypercubic lattice, which admit at least one Gibbs measure as a time-stationary measure. Under some mild non-degeneracy conditions on the rates and the specification, we prove that one can use the relative entropy as a Lyapunov functional and thereby show that all possible limit points of the associated measure-valued dynamics are also Gibbs measures if the initial condition is translation-invariant. In the special case of reversible interacting particle systems on one- and two-dimensional lattices, we extend the method to non-translation invariant initial distributions and completely determine the phase portrait of the dynamics. This answers open questions on the possibility of time-periodic behaviour in two dimensions and on the connection between uniqueness of the invariant measure and ergodicity. This is based on joint work with Benedikt Jahnel.

About Jonas

Jonas is a second year PhD Student at the Weierstraß Institute (WIAS) in the research group Probabilistic Methods for Dynamic Communication Networks under the supervision of Benedikt Jahnel and a Phase II Student at the Berlin Mathematical School.

Previously, he was a scholarship holder in the research group Data-driven Modeling of Complex Systems at FU Berlin, under the supervision of Péter Koltai and Gary Froyland.

He is co-founder of BLISS, a student-run club connecting interested students with AI/ML researchers and practicioners through weekly reading groups, invited talks, and coding challenges. If you want to know more about it or join one of their weekly meetings, write him an email!