Path-integrals and optimal paths for the fractional Ornstein–Uhlenbeck process

Abstract We derive the path-integral representation of the fractional Ornstein–Uhlenbeck process driven by nonstationary Riemann–Liouville fractional Gaussian noise (fGn), for both the subdiffusive and superdiffusive regimes. We express the corresponding action, which is a quadratic functional of individual trajectories of the process, in two alternative but equivalent forms: either in terms of fractional integrals or as a double integral with a nonlocal kernel. Moreover, we determine in closed form the optimal (action-minimizing) paths conditioned to reach a prescribed point at a fixed time moment and discuss their behavior, which appears to be non-intuitive for subdiffusive processes in the presence of a strong confining potential. We also demonstrate that such a behavior persists in case of a stationary fGn in the sense of Mandelbrot and van Ness.