Large deviations for the empirical measure of heavy-tailed Markov renewal processes

Abstract

A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker-Varadhan analysis of the problem.

Cycle symmetries and circulation fluctuations for discrete-time and continuous-time Markov chains

“…The following lemma, which is due to Mariani and Zambotti [25], shows that the empirical flow of Markov renewal processes satisfies a large deviation principle with a good rate function. Proof.…”

Random walks on quasi one dimensional lattices: Large deviations and fluctuation theorems

“…By the implicit function theorem and Lemma 5.1, the function (0, 1/α + ) ∋ ϑ → λ + (ϑ) ∈ (−∞, λ c ) is smooth. In particular, using (36), I + is smooth on (0, 1/α + ) where it holds…”

“…This condition is not satisfied when considering Markov random walks on quasi 1d lattices, hence in our case the results of [13], an the similar ones of [41], cannot be applied. In the context of LDPs for processes under random time changes we also mention the new progresses obtained in [33,36]. Restricting to the case w i ∈ {−1, 1} (which covers the applications to molecular motors), the process (Z t ) t∈R + becomes a random walk on Z with generic holding times (not necessarily exponential).…”

Large deviations of the empirical flow for continuous time Markov chains

“…In [29,30] LDPs for the current of the Brownian motion on a compact Riemann manifold are obtained. We mention also the recent preprint [35] on the joint large deviations for the empirical measure and flow for a renewal process on a finite graph. Currents play also a crucial role in biochemical processes, and the study of large fluctuations and related symmetries have recently received much attention (see e.g.…”