Residence Time Distribution for a Class of Gaussian Markov Processes

TL;DR

This paper investigates the residence time distribution of a family of Gaussian Markov processes, revealing a qualitative change in shape with increasing persistence exponent and providing exact recursive methods for moments and special cases.

Contribution

It introduces two recursive methods to calculate moments of the residence time distribution for Gaussian Markov processes and derives closed-form solutions for specific parameter values.

Findings

Residence time distribution shape changes qualitatively with persistence exponent.

Exact recursive formulas for moments of the distribution are developed.

Closed-form expressions are obtained for special parameter values.

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 $\alpha$. The persistence exponent for these processes is simply given by $\theta=\alpha$ but the residence time distribution is nontrivial. The shape of this distribution undergoes a qualitative change as $\theta$ 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 $\alpha$. For some special values of $\alpha$, we obtain closed form expressions of the distribution function.