A study of path measures based on second-order Hamilton--Jacobi equations and their applications in stochastic thermodynamics

TL;DR

This paper systematically explores the mathematical structure of path measures using second-order Hamilton--Jacobi equations, linking stochastic dynamics, measure theory, and thermodynamics, and deriving key principles like large deviation and entropy production.

Contribution

It introduces a unified framework connecting second-order HJ equations with stochastic measures, large deviations, and entropy in thermodynamics, advancing the theoretical understanding of stochastic systems.

Findings

Derivation of large deviation rate function from path measures.

Equivalence of rate function to Onsager--Machlup functional.

Decomposition of entropy production revealing thermodynamic irreversibility.

Abstract

This paper provides a systematic investigation of the mathematical structure of path measures and their profound connections to stochastic differential equations (SDEs) through the framework of second-order Hamilton--Jacobi (HJ) equations. This approach establishes a unified methodology for analyzing large deviation principles (LDPs), entropy minimization, and entropy production in stochastic systems. Second-order HJ equations are shown to play a central role in bridging stochastic dynamics and measure theory while forming the foundation of stochastic geometric mechanics and their applications in stochastic thermodynamics. The large deviation rate function is rigorously derived from the probabilistic structure of path measures and proved to be equivalent to the Onsager--Machlup functional of stochastic gradient systems coupled with second-order HJ equations. We revisit entropy minimization problems, including finite time horizon problems and Schr\"{o}dinger's problem, demonstrating the connections with stochastic geometric mechanics. Furthermore, we present a novel decomposition of entropy production for stochastic systems, revealing that thermodynamic irreversibility can be interpreted as the difference of the corresponding forward and backward second-order HJ equations. Together, this work establishes a comprehensive mathematical study of the relations between path measures and stochastic dynamical systems, and their diverse applications in stochastic thermodynamics and beyond.