Continuous-time multi-type Ehrenfest model and related Ornstein-Uhlenbeck diffusion on a star graph

• Main Authors: Di Crescenzo, Antonio, Martinucci, Barbara, Spina, Serena
• Format: Journal Article
• Language: English
• Published: 11-01-2022
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2201.03977

We deal with a continuous-time Ehrenfest model defined over an extended star
graph, defined as a lattice formed by the integers of $d$ semiaxis joined at
the origin. The dynamics on each ray are regulated by linear transition rates,
whereas the switching among rays at the origin occurs according to a general
stochastic matrix. We perform a detailed investigation of the transient and
asymptotic behavior of this process. We also obtain a diffusive approximation
of the considered model, which leads to an Ornstein-Uhlenbeck diffusion process
over a domain formed by semiaxis joined at the origin, named spider. We show
that the approximating process possesses a truncated Gaussian stationary
density. Finally, the goodness of the approximation is discussed through
comparison of stationary distributions, means and variances.