  In this article, we investigate the ergodic behaviour of a multidimensional
age-dependent branching process with a singular jump kernel, motivated by
studying the phenomenon of telomere shortening in cell populations. Our model
tracks individuals evolving within a continuous-time framework indexed by a
binary tree, characterised by age and a multidimensional trait. Branching
events occur with rates dependent on age, where offspring inherit traits from
their parent with random increase or decrease in some coordinates, while the
most of them are left unchanged. Exponential ergodicity is obtained at the cost
of an exponential normalisation, despite the fact that we have an unbounded
age-dependent birth rate that may depend on the multidimensional trait, and a
non-compact transition kernel. These two difficulties are respectively treated
by stochastically comparing our model to Bellman-Harris processes, and by using
a weak form of a Harnack inequality. We conclude this study by giving examples
where the assumptions of our main result are verified.
