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

2025/08/04 by Jianyu Hu, Hu, Jianyu, Qiao Huang +4
Physics and Astronomy · Engineering · #Advanced Thermodynamics and Statistical Mechanics #Statistical Mechanics and Entropy #Control and Stability of Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2508.02469

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 Schrödinger'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.

Citations

• A global structure-preserving kernel method for the learning of Poisson systems
• Entropy Production in Non-Gaussian Active Matter: A Unified Fluctuation Theorem and Deep Learning Framework
• Emergence of Newtonian Deterministic Causality from Stochastic Motions in Continuous Space and Time
• A Structure-Preserving Kernel Method for Learning Hamiltonian Systems
• Deep learning probability flows and entropy production rates in active matter
• Stochastic Geometric Mechanics in Nonequilibrium Thermodynamics: Schrödinger meets Onsager
• Data-driven method to learn the most probable transition pathway and stochastic differential equation
• The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit
• The Most Probable Transition Paths of Stochastic Dynamical Systems: A Sufficient and Necessary Characterization
• Information Projection on Banach spaces with Applications to State\n Independent KL-Weighted Optimal Control
• The Graph Limit of The Minimizer of The Onsager-Machlup Functional and Its Computation
• Gaussian approximations for transition paths in Brownian dynamics
• Stochastic derivatives and generalized h-transforms of Markov processes
• From the Schrödinger problem to the Monge–Kantorovich problem
• Stochastic Hamilton–Jacobi–Bellman Equations
• Conjugate convex functions in optimal stochastic control

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