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

If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting "MRCA age" process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright--Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on $\mathbb{R}_{++}\times\mathbb{R}_{++}$ with intensity measures of the form $\lambda\otimes\mu$ where $\lambda$ is Lebesgue 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 construction describe the time evolution of the MRCA age in $(1+\beta)$-stable continuous state branching processes conditioned on nonextinction--a particular case of which, $\beta=1$, is Feller's continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist.