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

## TL;DR

This paper analytically derives the two-time correlation function and occupation time distribution for tied-down renewal processes, generalizing the Brownian bridge, with implications for physical models and surface evolution.

## Contribution

It provides the first exact analytical expressions for correlation functions and occupation times in tied-down renewal processes, extending understanding beyond Brownian bridges.

## Key findings

- Exact asymptotic correlation expressions in different regimes
- Exact distribution of occupation times
- Implications for physical and surface evolution models

## Abstract

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 and in the persistence regime of large separations. We also investigate other quantities, such as the backward and forward recurrence times, as well as the occupation time of the process. The latter has a broad distribution which is 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.