Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates

TL;DR

This paper develops a comprehensive large deviation framework for one-dimensional run-and-tumble processes with space-dependent velocities and switching rates, analyzing their steady states and fluctuations at multiple levels.

Contribution

It introduces a detailed large deviations analysis at Level 2.5 for these processes, including methods to derive deviations of densities, currents, flows, and switching intervals.

Findings

Large deviations at Level 2.5 characterize fluctuations of empirical densities, currents, and switching flows.

Derived large deviations for empirical intervals between switching events using an auxiliary Markov chain.

Provided methods to obtain large deviations of any time-additive observable via contraction or deformed generator techniques.

Abstract

One-dimensional run-and-tumble processes may converge towards some localized non-equilibrium steady state when the two velocities and/or the two switching rates are space-dependent. A long dynamical trajectory can be then analyzed via the large deviations at Level 2.5 for the joint probability of the empirical densities, of the empirical spatial currents and of the empirical switching flows. The Level 2 for the empirical densities alone can be then derived via the optimization of the Level 2.5 over the empirical flows. More generally, the large deviations of any time-additive observable can be also obtained via contraction from the Level 2.5, or equivalently via the deformed generator method and the corresponding Doob conditioned process. Finally, the large deviations for the empirical intervals between consecutive switching events can be obtained via the introduction of the alternate Markov chain that governs the series of all the switching events of a long trajectory.