Markov trajectories: Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators

International audienceThe Ensemble of trajectories $x(0 ≤ t ≤ T)$ produced by the Markov generator $M$ in a discrete configuration space can be considered as ’Canonical’ for the following reasons : (C1) the probability of the trajectory $x(0 ≤ t ≤ T)$ can be rewritten as the exponential of a linear combination of its relevant empirical time-averaged observables $E_n$, where the coefficients involving the Markov generator are their fixed conjugate parameters; (C2) the large deviations properties of these empirical observables En for large $T$ are governed by the explicit rate function $I^{[2.5]}_M$ $(E.)$ at Level 2.5, while in the thermodynamic limit $T = +∞$, they concentrate on their typical values $E^{typ[M]}_n$ determined bythe Markov generator $M$. This concentration property in the thermodynamic limit $T = +∞$ suggests to introduce the notion of the ’Microcanonical Ensemble’ at Level 2.5 for stochastic trajectories $x(0 ≤ t ≤ T)$, where all the relevant empirical variables $E_n$ are fixed to some values $E*_n$n and cannot fluctuate anymore for finite $T$. The goal of the present paper is to discuss its main properties : (MC1) when the long trajectory $x(0 ≤ t ≤ T)$ belongs the Microcanonical Ensemble with the fixed empirical observables $E*_n$, the statistics of its subtrajectory $x(0 ≤ t ≤ τ )$ for 1$\ll \tau \ll T$ is governed by the Canonical Ensemble associated to the Markov generator $M*$ that would make the empirical observables $E*_n$ typical ; (MC2) in the Microcanonical Ensemble, the central role is played by the number \omega ^{[2.5]}_T$(E*)$ of stochastic trajectories of duration $T$ with the given empirical observables $E*_n$, and by the corresponding explicit Boltzmann entropy S$^{[2.5]}$$(E*)$ = $[ln \omega^{[2.5]}_T$$(E*)]$]/$T$. This general framework is applied to continuous-time Markov Jump processes and to discrete-time Markov chainswith illustrative example