The genealogy of continuous-state branching processes with immigration (Q1862498)

J. F. Le Gall and Y. Le Jan have extended the genealogical structure of Galton-Watson processes to continuous state branching processes via a non-Markovian process called the height process. In this paper, continuous state branching processes with immigration are studied. The height process \(H\) is defined as a simple local time functional of a strong Markov process \(X^*\), called the genealogy-coding process. There are given a pathwise construction of \(X^*\) based on a Lévy process \(X\) with nonnegative jumps that does not drift to \(+\infty\) and whose Laplace exponent coincides with the branching mechanism, and an independent subordinator \(Y\) whose Laplace exponent coincides with the immigration mechanism. The local time process \(H\) is found to be a continuous state branching process with immigration. An analog of the classical Ray-Knight-Williams theorem for a general Lévy process with no negative jumps is derived.