Large deviations for dynamical fluctuations of open Markov processes, with application to random cascades on trees

The large deviations at 'Level 2.5 in time' for time-dependent ensemble-empirical-observables, introduced by Maes et al (2008 Markov Proc. Relat. Fields 14 445) for the case of 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'. (paper)