Ray-Knight compactification of birth and death processes

A birth and death process is a continuous-time Markov chain with the minimal state space $\mathbb N$, whose transition matrix is standard and whose density matrix is the given birth-death matrix. Birth and death process is unique if and only if $\infty$ is an entrance or natural. When $\infty$ is neither an entrance nor natural, there are two ways in the literature to obtain all birth and death processes. The first one is an analytic treatment proposed by Feller in 1959, and the second one is a probabilistic construction completed by Wang in 1958. In this paper we will give another way to study birth and death processes using the Ray-Knight compactification. This way has the advantage of both the analytic and probabilistic treatments above. By applying the Ray-Knight compactification, every birth and death process can be modified into a c\`adl\`ag Ray process on $\mathbb N\cup \{\infty\}\cup\{\partial\}$, which is either a Doob processes or a Feller $Q$-process. Every birth and death process in the second class has a modification that is a Feller process on $\mathbb N\cup\{\infty\}\cup \{\partial\}$. We will derive the expression of its infinitesimal generator, which explains its boundary behaviours at $\infty$. Furthermore, by utilizing transformations of killing and Ikeda-Nagasawa-Watanabe's piecing out, we will also provide a probabilistic construction for birth and death processes. This construction relies on a triple determining the resolvent matrix introduced by Wang and Yang in their work (1992')}.