Stochastic Calculus for Pathwise Observables of Markov-Jump Processes: Unification of Diffusion and Jump Dynamics

TL;DR

This paper develops a unified stochastic calculus framework for path-wise observables in Markov-jump processes, paralleling diffusion process theories, enabling comprehensive thermodynamic analysis of jump and diffusion dynamics.

Contribution

It introduces a complete stochastic calculus for jump processes, unifying diffusion and jump dynamics, and connects classical and quantum descriptions of thermal systems.

Findings

Formulated a Langevin equation for jump processes.

Established covariation structure for path-wise observables.

Unified diffusion and jump dynamics through continuum limit.

Abstract

Path-wise observables--functionals of stochastic trajectories--are at the heart of time-average statistical mechanics and are central to thermodynamic inequalities such as uncertainty relations, speed limits, and correlation-bounds. They provide a means of thermodynamic inference in the typical situation, when not all dissipative degrees of freedom in a system are experimentally accessible. So far, theories focusing on path-wise observables have been developing in two major directions, diffusion processes and Markov-jump dynamics, in a virtually disjoint manner. Moreover, even the respective results for diffusion and jump dynamics were derived with a patchwork of different approaches that are predominantly indirect. Stochastic calculus was recently shown to provide a direct approach to path-wise observables of diffusion processes, while a corresponding framework for jump dynamics remained elusive. In our work we develop, in an exact parallelism with continuous-space diffusion, a complete stochastic calculus for path-wise observables of Markov-jump processes. We formulate a "Langevin equation" for jump processes, define general path-wise observables, and establish their covariation structure, whereby we fully account for transients and time-inhomogeneous dynamics. We prove the known kinds of thermodynamic inequalities in their most general form and discus saturation conditions. We determine the response of path-wise observables to general (incl. thermal) perturbations and introduce a corresponding response-function formalism. We carry out the continuum limit to achieve the complete unification of diffusion and jump dynamics. In addition, we connect the framework to quantum unraveling and the Belavkin equation for open quantum systems, associating quantum and classical descriptions of thermal systems.