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

Abstract

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.