High-accuracy discrete path integral solutions for stochastic processes with noninvertible diffusion matrices

Abstract

The derivation of the discrete path integral solution for the propagator is known to present a special problem for those stochastic processes whose diffusion matrices are noninvertible. In this paper two methods for formulating the stochastic dynamics in terms of path integrals are developed that are applicable whether or not the diffusion matrix is invertible. One of the methods is an extension of the standard technique available for the derivation of the functional formalism from Langevin equations. An accurate discretization scheme is used to replace these equations by finite-difference equations and a short time approximation for the propagator is then derived in terms of known statistical properties of noise terms. An alternative derivation of the discrete path integral is presented in terms of the Fokker-Planck formulation without the necessity of introducing discretization schemes into the discussion. This is achieved by making use of the cumulant generating function which is different in this realm. The mutual correspondence of the methods is established and their possible extensions are discussed. Both methods are indeed rigorous and allow for the systematic derivation of the short time propagator valid to any desired precision in a time increment \ensuremath{\tau}. Its use in a path integral means a significant reduction of the number of time steps that are required to achieve a given level of accuracy for a given net increment t=N\ensuremath{\tau}, and, therefore, significantly increasing the feasibility of path integral calculations. Another attractive feature of the present techniques is that they permit the efficient treatment of equations with singular diffusion matrices, two of which, a Kramers equation and a colored-noise problem, are considered.