Microcanonical conditioning of Markov processes on time-additive observables

TL;DR

This paper develops a comprehensive framework for microcanonical conditioning of various Markov processes on time-additive observables, extending previous work and connecting to large deviation theory.

Contribution

It introduces a general formalism for microcanonical conditioning of Markov processes on time-additive observables across different process types.

Findings

Framework applicable to discrete and continuous Markov processes

Explicit examples illustrating the formalism

Connection established with large deviation analysis

Abstract

The recent study by B. De Bruyne, S. N. Majumdar, H. Orland and G. Schehr [arXiv:2110.07573], concerning the conditioning of the Brownian motion and of random walks on global dynamical constraints over a finite time-window $T$, is reformulated as a general framework for the 'microcanonical conditioning' of Markov processes on time-additive observables. This formalism is applied to various types of Markov processes, namely discrete-time Markov chains, continuous-time Markov jump processes and diffusion processes in arbitrary dimension. In each setting, the time-additive observable is also fully general, i.e. it can involve both the time spent in each configuration and the elementary increments of the Markov process. The various cases are illustrated via simple explicit examples. Finally, we describe the link with the 'canonical conditioning' based on the generating function of the time-additive observable for finite time $T$, while the regime of large time $T$ allows to recover the standard large deviation analysis of time-additive observables via the deformed Markov operator approach.