Variational principles and Lagrangian functions for stochastic processes and their dissipative statistical descriptions

A variational Lagrangian formulation for stochastic processes and for the evolution equations of the associated probability density functions is developed. Particular attention is dedicated to Poisson-Kac processes possessing finite propagation velocity. The variational formulation in terms of Lagrangian and Hamiltonian densities permits to address different forms of “reversibility” characterizing these processes with respect to processes driven by nowhere differentiable Wiener fluctuations, and associated with the concepts of dynamic and statistical reversibility. The latter property, i.e. the statistical reversibility, implies that the extended Markov operator associated with Poisson-Kac and Generalized Poisson-Kac processes forms a group, continuously parametrized with respect to time.