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

• Published in: Mathematical methods in the applied sciences Vol. 45; no. 9; pp. 5483 - 5512
• Main Authors: Di Crescenzo, Antonio, Martinucci, Barbara, Spina, Serena
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01-06-2022
• Subjects: Approximation; Asymptotic properties; branching processes; diffusion processes; Ehrenfest model; Mathematical analysis; Ornstein–Uhlenbeck process; stationary distribution
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8123

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.