For $\alpha > 0$, the $\alpha$-Lipschitz minorant of a function $f:
\mathbb{R} \to \mathbb{R}$ is the greatest function $m : \mathbb{R} \to
\mathbb{R}$ such that $m \leq f$ and $|m(s)-m(t)| \le \alpha |s-t|$ for all
$s,t \in \mathbb{R}$, should such a function exist. If $X=(X_t)_{t \in
\mathbb{R}}$ is a real-valued L\'evy process that is not pure linear drift with
slope $\pm \alpha$, then the sample paths of $X$ have an $\alpha$-Lipschitz
minorant almost surely if and only if $| \mathbb{E}[X_1] | < \alpha$.

We consider stochastic equations of the form $X_k = \phi_k(X_{k+1}) Z_k$, $k
\in \mathbb{N}$, where $X_k$ and $Z_k$ are random variables taking values in a
compact group $G_k$, $\phi_k: G_{k+1} \to G_k$ is a continuous homomorphism,
and the noise $(Z_k)_{k \in \mathbb{N}}$ is a sequence of independent random
variables.

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.