Level 2.5 large deviations and uncertainty relations for self-interacting jump processes: tilting constructions and the emergence of time-scale separation

TL;DR

This paper develops a large deviation framework for self-interacting jump processes, revealing time-scale separation and extending uncertainty relations to non-Markovian systems, with applications to kinetic and thermodynamic bounds.

Contribution

It introduces a level-2.5 large deviation principle for SIJPs, utilizing tilting constructions to analyze non-Markovian dynamics and derive extended uncertainty relations.

Findings

Separation of microscopic and memory-driven timescales in SIJPs

Extension of uncertainty relations to non-Markovian processes

Validation through simple illustrative examples

Abstract

Self-interacting jump processes (SIJPs) describe systems with non-Markovian stochastic dynamics in which transition rates depend on empirical observables of the process, which gives rise to long-range memory and feedback. We derive the ``level-2.5'' large deviation (LD) principle governing the joint fluctuations of empirical occupation measure and the flux matrix for a broad class of SIJPs with general functional dependence on an empirical observable. The derivation is based on an exponential tilting construction and reveals a separation between a faster timescale of the microscopic dynamics and a slower timescale of the memory-driven evolution of transition rates, which is expressed through an exponentially discounted LD rate functional. Using this variational framework, we derive kinetic and thermodynamic uncertainty relations that extend classical Markovian bounds to non-Markovian systems, and illustrate their performance with simple examples.