Large Deviations for dynamical fluctuations of Open Markov processes, with application to random cascades on trees

TL;DR

This paper extends large deviation principles at 'Level 2.5' to open Markov processes with reservoirs, providing explicit formulas and applying the framework to analyze random cascades on trees with particle injection and removal.

Contribution

It introduces a formalism for large deviations in open Markov processes with time-dependent rules and applies it to complex tree cascade models.

Findings

Explicit formulas for joint probabilities of occupation numbers and flows

Extension of large deviation principles to open systems with reservoirs

Application to particle cascades on trees with source and sink reservoirs

Abstract

The large deviations at 'Level 2.5 in time' for time-dependent ensemble-empirical-observables, introduced by C. Maes, K. Netocny and B. Wynants [Markov Proc. Rel. Fields. 14, 445 (2008)] for the case of $N$ independent Markov jump processes, are extended to the case of open Markov processes with reservoirs : explicit formulas are given for the joint probability of empirical occupation numbers and empirical flows, both for discrete-time dynamics and for continuous-time jump dynamics, with possibly time-dependent dynamical rules and/or time-dependent driving of the reservoirs. This general formalism is then applied to random cascades on trees, where particles are injected at the root via a 'source reservoir', while the particles are removed at the leaves of the last generation of the tree via 'sink reservoirs'.