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

Lars Torbjørn Stutzer1,*, Cai Dieball1, and Aljaž Godec2,1,†

  • *Present address: Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany.
  • †Contact author: agodec@physik.uni-freiburg.de
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Phys. Rev. X 16, 021038 – Published 19 May, 2026

DOI: https://doi.org/10.1103/9ncx-4pgr

Abstract

Pathwise observables—functionals of stochastic trajectories—are at the heart of time-averaged 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 pathwise 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 pathwise 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 pathwise observables of Markov-jump processes. We formulate a “Langevin equation” for jump processes, define general pathwise 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 discuss saturation conditions. We determine the response of pathwise observables to general (including 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. Our results open avenues in the direction of discrete-state analogs of generative diffusion models and the learning of stochastic thermodynamics from fluctuating trajectories.

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We develop, in an exact parallelism with the results for diffusion processes, a stochastic calculus for pathwise observables of general Markov-jump processes. To demonstrate the power of the approach, we prove the known kinds of thermodynamic inequalities in their most general form, discuss their saturation, and determine the response of pathwise observables to general perturbations. Finally, we establish the continuum limit to achieve the complete unification of diffusion and jump dynamics. Our results place functionals of diffusion and jump processes on an equal footing on the level of individual realizations, and hence achieve a “contraction” of the two until now disjoint frameworks. The methodological advance opens new avenues in the direction of discrete-state analogs of generative diffusion models and the learning of stochastic thermodynamics.

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Phys. Rev. X 16, 021038– Published 19 May, 2026
  • Received 2 August 2025
  • Accepted 17 April 2026
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