Residence time distribution for a class of Gaussian Markov processes

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