Covariant lagrangian of Onsager and Machlup without discretization

A well defined notion of generalized Onsager-Machlup Lagrangian in the covariant formalism of general diffusion processes is obtained without any discretization procedure. Main ideas are to clarify some basic links between the conventional Fokker-Planck description and a Schrödingerlike description, and to exploit afterwards the well established path integral formalism of quantum mechanics. The first step is realized with the help of a modified version of Nelson's stochastic mechanics, that is, “thermal mechanics”, which seems well adapted to nonequilibrium statistical thermodynamics. A natural notion of deterministic approximation for the general diffusion process is also obtained in the present framework.