Stochastic control and nonequilibrium thermodynamical systems

Minus the logarithm of the density of a diffusion process is shown to be the value function of a stochastic control problem, where the controlled equation evolves backward in time. For nonequilibrium thermodynamical systems, this provides a Hamilton-Jacobi-like theory, where the action is a local entropy function. This variational principle may also be seen as a rigorous version of the formal Onsager-Machlup principle. For the Ornstein-Uhlenbeck model of physical Brownian motion, the principle is related to a pathwise Newton law. For the latter model, several other pathwise results are derived, which strengthen the classical thermodynamical results on the averages. In particular, the (stochastic) Helmholtz free energy is shown to be a backward submartingale with respect to the natural filtration.