Hamiltonian systems with a stochastic force:nonlinear versus linear, and a girsanov formula

Abstract

Abstract We discuss stochastic perturbations of classical Hamiltonian systems by a white noise force. We prove existence and uniqueness results for the solutions of the equation of motion under general conditions on the classical system, as well as their continuous dependence on the initial conditions. We also prove that the process in phase space is a diffusion with transition probability densities, and Lebesgue measure as c-finite invariant measure. We prove a Girsanov formula relating the solution for a nonlinear force with the one for a linear force, and give asymptotic estimates on functions of the phase space process Keywords: Stochastic Hamiltonian systemssecond order Ito processesnon linear stochastic differential equationsGirsanov formulaasymptotic estimates for momentsσ-finite invariant measurehypoelliptic generators †CERFIM, Locarno(CH) *SFB-237 Bochum-Essen-Duússeldorf †CERFIM, Locarno(CH) *SFB-237 Bochum-Essen-Duússeldorf Notes †CERFIM, Locarno(CH) *SFB-237 Bochum-Essen-Duússeldorf