Occupation time of a renewal process coupled to a discrete Markov chain

Abstract

A semi-Markov process is one that changes states in accordance with a Markov chain but takes a random amount of time between changes. We consider the generalisation to semi-Markov processes of the classical Lamperti law for the occupation time of a two-state Markov process. We provide an explicit expression in Laplace space for the distribution of an arbitrary linear combination of the occupation times in the various states of the process. We discuss several consequences of this result. In particular, we infer the limiting distribution of this quantity rescaled by time in the long-time scaling regime, as well as the finite-time corrections to its moments.

arXiv comment18 pages, 4 figures

References in corpus (23)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Nontrivial Exponent for Simple Diffusion
• Persistent spins in the linear diffusion approximation of phase ordering and zeros of stationary gaussian processes
• Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
• Weakly non-ergodic Statistical Physics
• Large deviations and nontrivial exponents in coarsening systems
• Diffusive persistence and the `sign-time' distribution
• Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process
• Lamperti-type laws
• Residence Time Distribution for a Class of Gaussian Markov Processes
• Generalized persistence exponents: an exactly soluble model
• Diffusions conditioned on occupation measures
• Sign-time distributions for interface growth
• Statistics of persistent events in the binomial random walk: Will the drunken sailor hit the sober man?
• Renewal theory with fat tailed distributed sojourn times: typical versus rare
• Stationary definition of persistence for finite temperature phase ordering
• Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos
• Statistics of occupation time and connection to local properties of non-homogeneous random walks
• Statistics of the occupation time for a class of Gaussian Markov processes
• Temporal Correlations and Persistence in the Kinetic Ising Model: the Role of Temperature
• Two-time correlation and occupation time for the Brownian bridge and tied-down renewal processes
• Statistics of the occupation time for a random walk in the presence of a moving boundary
• Occupation time of a randomly accelerated particle on the positive half axis: Results for the first five moments