Evaluation of the probability current in the stochastic path integral formalism

The probability current is a vital quantity in the Fokker-Planck description of stochastic processes. It characterizes non-equilibrium stationary states and appears in linear response calculations. We recover and review the probability current in the Onsager-Machlup functional approach to Markov processes by deriving a self-contained expression in general non-equilibrium fluctuation-dissipation relations using field theoretical methods. The derived formulas hold for non-constant drift and diffusion tensors and are explicitly evaluated in an Ornstein-Uhlenbeck process with non-reciprocal interactions specified as a harmonically bound particle in shear flow. Our work clarifies the concept of the probability current – familiar from the Fokker-Planck equation – in the path integral approach.