Markov Processes with Restart

TL;DR

This paper studies Markov processes with random restarts, providing explicit formulas for their invariant measures and moments, and demonstrating their ergodic properties, with applications to Brownian motions and mobility modeling.

Contribution

It introduces a general framework for Markov processes with restarts, deriving explicit formulas and ergodic properties, expanding understanding of such processes in various applications.

Findings

Explicit formulas for invariant measures and moments.

The process is always positive Harris recurrent.

The process is exponentially ergodic with rate ≥ restart rate.

Abstract

We consider a general honest homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such processes comes from modeling human and animal mobility patterns, restart processes in communication protocols, and from application of restarting random walks in information retrieval. We provide a connection between the transition probability functions of the original Markov process and the modified process with restarts. We give closed-form expressions for the invariant probability measure of the modified process. When the process evolves on the Euclidean space there is also a closed-form expression for the moments of the modified process. We show that the modified process is always positive Harris recurrent and exponentially ergodic with the index equal to (or bigger than) the rate of restarts. Finally, we illustrate the general results by the standard and geometric Brownian motions.