Statistics of the occupation time for a class of Gaussian Markov processes

Abstract

We revisit the work of Dhar and Majumdar [Phys. Rev. E 59, 6413 (1999)] on the limiting distribution of the temporal mean $M_{t}=t^{-1}\int_{0}^{t}du \sign y_{u}$, for a Gaussian Markovian process $y_{t}$ depending on a parameter $α$, which can be interpreted as Brownian motion in the scale of time $t^{\prime}=t^{2α}$. This quantity, for short the mean `magnetization', is simply related to the occupation time of the process, that is the length of time spent on one side of the origin up to time t. Using the fact that the intervals between sign changes of the process form a renewal process in the time scale t', we determine recursively the moments of the mean magnetization. We also find an integral equation for the distribution of $M_{t}$. This allows a local analysis of this distribution in the persistence region $(M_t\to\pm1)$, as well as its asymptotic analysis in the regime where $α$ is large. We finally put the results thus found in perspective with those obtained by Dhar and Majumdar by another method, based on a formalism due to Kac.