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

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, entropy production, and inverse learning problems in stochastic systems. The 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 demonstrated 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. In the context of stochastic thermodynamics, we present a novel decomposition of entropy production, revealing that thermodynamic irreversibility can be interpreted as the difference of the corresponding forward and backward second-order HJ equations. Furthermore, we tackle the challenging problem of identifying stochastic gradient systems from observed most probable paths by reformulating the original nonlinear and non-convex problem into a linear and convex framework through a second-order HJ equation. 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.