Stochastic deformation of integrable dynamical systems and random time symmetry

Abstract

We present a deformation of a class of elementary classical integrable systems using stochastic diffusion processes. This deformation applies to the solution of the associated classical Newtonian, Hamiltonian, Lagrangian, and variational problems and to the Hamilton–Jacobi method of characteristics. The underlying stochastic action functionals involve dual random times, whose expectations are connected to the new variables of the system after a canonical transformation.