Canonical stochastic dynamical systems

Abstract

A canonical stochastic dynamical system for time-symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations. A certain class of stochastic dynamical systems gives a Hamiltonian formalism of Nelson’s stochastic mechanics. In a manner analogous to classical mechanics, the notions of a stochastic Poisson bracket and canonical transformation are introduced to the stochastic dynamical systems. It is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation.