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If we follow an asexually reproducing population through time, then theamount of time that has passed since the most recent common ancestor (MRCA) ofall current individuals lived will change as time progresses. The resulting"MRCA age" process has been studied previously when the population has aconstant large size and evolves via the diffusion limit of standardWright--Fisher dynamics. For any population model, the sample paths of the MRCAage process are made up of periods of linear upward drift with slope +1punctuated by downward jumps. We build other Markov processes that have suchpaths from Poisson point processes on $\mathbb{R}_{++}\times\mathbb{R}_{++}$with intensity measures of the form $\lambda\otimes\mu$ where $\lambda$ isLebesgue measure, and $\mu$ (the "family lifetime measure") is an arbitrary,absolutely continuous measure satisfying $\mu((0,\infty))=\infty$ and$\mu((x,\infty))<\infty$ for all $x>0$. Special cases of this constructiondescribe the time evolution of the MRCA age in $(1+\beta)$-stable continuousstate branching processes conditioned on nonextinction--a particular case ofwhich, $\beta=1$, is Feller's continuous state branching process conditioned onnonextinction. As well as the continuous time process, we also consider thediscrete time Markov chain that records the value of the continuous processjust before and after its successive jumps. We find transition probabilitiesfor both the continuous and discrete time processes, determine when theseprocesses are transient and recurrent and compute stationary distributions whenthey exist.

Dynamics of the time to the most recent common ancestor in a large branching population