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Author: Renger, D.R. Michiel × End date: 2019 × Start date: 2010 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: large deviations ×

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    On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Peletier, Mark A.; Renger, D.R. Michiel
    Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions L that induce a flow, given by L(pt, pt) = 0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when L is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.
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    Large deviations of specific empirical fluxes of independent Markov chains, with implications for Macroscopic Fluctuation Theory
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Renger, D.R. Michiel
    We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.