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

We revisit the work of Dhar and Majumdar (1999 Phys. Rev. E 59 6413) on the limiting distribution of the temporal mean M-t = t(-1) integral (t)(0) du sign y(u), for a Gaussian Markovian process y(t) depending on a parameter a , which can be interpreted as Brownian motion in the time scale t' = t(2 al pha). This quantity, the mean 'magnetization', is simply related to the occ upation time of the process, that is the length of time spent on one side o f the origin up to time t. Using the fact that the intervals between sign c hanges of the process form a renewal process on the time scale t', we deter mine recursively the moments of the mean magnetization. We also find an int egral equation for the distribution of M-t. This allows a local analysis of this distribution in the persistence region (M-t --> +/-1), as well as its asymptotic analysis in the regime where alpha is large. Finally, we put th e results thus found in perspective with those obtained by Dhar and Majumda r by another method, based on a formalism due to Kac.