Occupation times via Bessel functions

TL;DR

This paper derives explicit formulas for occupation time densities in continuous-time Markov processes, using Fourier, Laplace transforms, and Bessel functions, with applications to molecular state fluctuation analysis.

Contribution

It introduces a novel method combining Fourier and Laplace transforms to compute occupation time densities for Markov processes with countably many states, including explicit formulas involving Bessel functions.

Findings

Derived explicit occupation time density for simple random walk on Z

Connected occupation times to modified Bessel functions via spectral measures

Provided analytical tools for studying state occupation in Markov processes

Abstract

This study of occupation time densities for continuous-time Markov processes was inspired by the work of E.Nir et al (2006) in the field of Single Molecule FRET spectroscopy. There, a single molecule fluctuates between two or more states, and the experimental observable depends on the state's occupation time distribution. To mathematically describe the observable there was a need to calculate a single state occupation time distribution. In this paper, we consider a Markov process with countably many states. In order to find a one-stete occupation time density, we use a combination of Fourier and Laplace transforms in the way that allows for inversion of the Fourier transform. We derive an explicit expression for an occupation time density in the case of a simple continuous time random walk on Z. Also we examine the spectral measures in Karlin-McGregor diagonalization in an attempt to represent occupation time densities via modified Bessel functions.