Energetics of Poisson-Kac stochastic processes possessing finite propagation velocity

A local fluctuation–dissipation theorem for the power delivered by a stochastic forcing is derived for Ornstein–Uhlenbeck processes driven by smooth, i. e. almost everywhere (a. e.)-differentiable stochastic perturbations (Poisson–Kac processes). An analytic expression for the probability density function of the fluctuational power is obtained in the large time limit. As these processes converge, in the Kac limit, toward classical Langevin equations driven by Wiener processes, a coarse-grained analysis of the statistical properties of the fluctuational work is developed.