Sojourn time in an union of intervals for diffusions

TL;DR

This paper introduces a method to compute the Laplace transform of the time a diffusion process spends in a union of intervals, with applications in biology and a special focus on Brownian motion.

Contribution

It presents a novel approach using differential equations to calculate the Laplace transform of sojourn times in unions of intervals for linear diffusions.

Findings

Method applicable to biological models of membrane receptor clustering

Explicit representation for Brownian motion's local time in finite sets

Provides a computational tool for analyzing diffusion-related sojourn times

Abstract

We give a method for computing the iterated Laplace transform of the sojourn time in an union of intervals for linear diffusion processes. This random variable comes from a model occurring in biology concerning the clustering of membrane receptors. The way used hinges on solving differential equations. We finally have a look on the particular case of Brownian motion and we provide a representation for the Laplace transform of its local time in a finite set.