Residence Time Distribution for a Class of Gaussian Markov Processes

Abstract

We study the distribution of residence time or equivalently that of ``mean magnetization" for a family of Gaussian Markov processes indexed by a positive parameter $α$. The persistence exponent for these processes is simply given by $θ=α$ but the residence time distribution is nontrivial. The shape of this distribution undergoes a qualitative change as $θ$ increases, indicating a sharp change in the ergodic properties of the process. We develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary $α$. For some special values of $α$, we obtain closed form expressions of the distribution function.