Two-time Stochastic Lagrangian Dynamics

This paper defines and Studies the two-time stochastic dynamical systems that are connected to two-time stochastic laws of motion. Section I formulates and studies the two-time stochastic flows on manifolds. Section 2 referes to the HU principle wich unifies the Hamiltonian and Lagarangian description of a dynamical system based on curvilinear integral actions. Our action integral consists of two path dependent curvilinear integrals and one path dependent Stratonovich Curvilinear integral. The stochastic extremals are solutions of two-time stochastic Euler-Lagrange-Pfaff equations, describing a geometrical distribution.