Path integrals and non-Markov processes. I. General formalism

Abstract

We develop the path-integral formalism as applied to non-Markov stochastic processes in order to allow us to study the effects of colored external noise on a physical system. The system we initially consider consists of a Langevin equation x\ifmmode \dot{}\else \.{}\fi{}=-V'(x)+\ensuremath{\xi}, where \ensuremath{\xi} is a Gaussian noise with zero mean and correlator 〈\ensuremath{\xi}(t)\ensuremath{\xi}(t')〉=(D/\ensuremath{\tau})C(\ensuremath{\Vert}t-t'\ensuremath{\Vert}/\ensuremath{\tau}), \ensuremath{\tau} being the noise correlation time. Starting from the Langevin equation, we obtain a path-integral representation for probability density functions on the infinite time interval -\ensuremath{\infty}<t<\ensuremath{\infty}, and show how in certain cases a simple representation also exists in terms of a sum over paths on a finite time interval. The weighting factor for paths in this latter case consists of an exponential factor which is a generalization of that originally found by Onsager and Machlup but also contains nontrivial boundary terms depending on the initial preparation of the system.