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

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 ẋ=-V’(x)+ξ, where ξ is a Gaussian noise with zero mean and correlator 〈ξ(t)ξ(t’)〉=(D/τ)C(‖t-t’‖/τ), τ 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 -∞<t<∞, 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.