This talk investigates the mathematical structure of path measures, both from a measure-theoretical perspective and through stochastic differential equations. The realization of path measures as Langevin systems hinges on the pivotal role of second-order Hamilton-Jacobi-Bellman equations, which form the foundation of stochastic geometric mechanics and applications in stochastic thermodynamics. We explore the emergence of the Onsager-Machlup functional in large deviation theory, the rates of entropy production in irreversible thermodynamic processes, and entropy minimization problems encoded in stochastic geometric mechanics.