Ergodicity of an infinite dimensional renewal process
A very special infinite-dimensional renewal process is constructed and studied. Processes of this type have been introduced by \textit{F. Spitzer} [Adv. Math., Suppl. Stud. 9, 147-155 (1986; Zbl 0608.60083)]. The state space of the process is \(\mathbb{R}^{\mathbb{Z}}_ +\). A state \(\eta=(\eta(k))_{k\in\mathbb{Z}}\) represents a vector of ages of objects. The \(k\)-th coordinate \(\eta(k)\) increases deterministically at rate 1 or jumps to 0 with failure rate \(c+{1\over 2}(\eta(k-1)+\eta(k+1))\) for some constant \(c\geq 0\). The main result states that this process is ergodic if and only if \(c>0\). The construction of the process requires a major part of the paper. It is modelled as a limit of Markov processes with finite- dimensional state spaces.
scientific article; zbMATH DE number 68094