Level-crossing counting for generalized Langevin equations.

Abstract

We address the counting of level crossings for the generalized Brownian motion in which the motion of a Brownian particle (unbound as well as linearly bound) is governed by an integrodifferential stochastic equation with a memory dissipation kernel: the generalized Langevin equation. We suppose that the driving noise is internal and the fluctuation-dissipation relation holds. The most common assumption is that the driving noise is Gaussian, which allows us to make a complete analytical treatment of the whole problem, including crossing statistics. We obtain the general asymptotic behavior for regular driving noises having finite intensity and fast decaying correlations, as well as for fractional noises with long-time tail correlations and slow decay, which are related to anomalous diffusion. For the generalized Brownian oscillator we study the stationary (equilibrium) state and the approach to equilibrium.