Path integral solutions for non-Markovian processes

For a nonlinear stochastic flow driven by Markovian or non-Markovian colored noise (t) we present the path integral solution for the single-event probabilityp(x,t). The solution has the structure of a complex-valued double path integral. Explicit formulas for the action functional, i.e., the non-Markovian Onsager-Machlup functional, are derived for the case that (t) is characterized by a stationary Gaussian process. Moreover, we derive explicit results for (generalized) Poissonian colored shot noise (t). The use of the path integral solution is elucidated by a weak noise analysis of the WKB-type. As a simple application, we consider stochastic bistability driven by colored noise with an extremely long correlation time.