Two-time correlation and occupation time for the Brownian bridge and tied-down renewal processes

Claude Godrèche

Published 2017-04-14Version 1

Tied-down renewal processes are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time $t$. We give an analytical derivation of the two-time correlation function for such processes in the Laplace space of all temporal variables. This yields the exact asymptotic expression of the correlation in the Porod regime of short separations between the two times. We also investigate other quantities, such as the backward and forward recurrence times, as well as the occupation time of the process, which are determined exactly. Physical implications of these results for the Poland Scheraga and related models are given. These results also give exact answers to questions posed in the past in the context of stochastically evolving surfaces.