A detailed discussion of the path-integral formalism for stochastic processes described by a stochastic differential equation driven by a nonwhite noise is given. The path-integral representation in the configuration space of the transition probability for a process driven by Ornstein-Uhlenbeck noise is derived. We show how to treat in this approach any kind of initial conditions, including the question of the coupling with the noise at initial time. Known approximations are reobtained in this context. Markovian approximations based on the Lagrangian are also discussed. The stationary distribution of the process in the weak-noise limit is obtained from the Lagrangian without relying on the use of Fokker-Planck or Markovian approximations.

• Received 26 June 1989

DOI:https://doi.org/10.1103/PhysRevA.40.7312