Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps

TL;DR

This paper analyzes large deviations in one-dimensional Jump-Drift and Jump-Diffusion processes, focusing on density, current, jump-flow, and excursions, providing a comprehensive probabilistic framework for long-term trajectory behaviors.

Contribution

It introduces a dual large deviations framework at Level 2.5 for both empirical measures and jump excursions, applied to various jump processes to reveal simplifications in rate functions.

Findings

Large deviations for empirical density, current, and jump-flow are characterized.

Explicit rate functions are derived for different jump processes.

Simplifications occur in rate functions for specific process classes.

Abstract

For one-dimensional Jump-Drift and Jump-Diffusion processes converging towards some steady state, the large deviations of a long dynamical trajectory are described from two perspectives. Firstly, the joint probability of the empirical time-averaged density, of the empirical time-averaged current and of the empirical time-averaged jump-flow are studied via the large deviations at Level 2.5. Secondly, the joint probability of the empirical jumps and of the empirical excursions between consecutive jumps are analyzed via the large deviations at Level 2.5 for the alternate Markov chain that governs the series of all the jump events of a long trajectory. These two general frameworks are then applied to three examples of positive jump-drift processes without diffusion, and to two examples of jump-diffusion processes, in order to illustrate various simplifications that may occur in rate functions and in contraction procedures.