  We study the long-time behaviour of the first-moment semigroup of a non
conservative piecewise deterministic measure-valued stochastic process with
support on R 2 + driven by a deterministic flow between random jump times, with
a transition kernel which has a degenerate form. Using a Doob h-transform where
the function h is taken as an eigenfunction of the associated generator, we can
bring ourselves back to the study of a conservative process whose exponential
ergodicity is proven via Harris' Theorem. Particular attention is given to the
proof of Doeblin minoration condition. The main difficulty is the degeneracy of
one of the two variables, and the deterministic dependency between the two
variables, which make it no trivial to uniformly bound the expected value of
the trajectories with respect to a non-degenerate measure in a two-dimensional
space, which is particularly hard in a non-compact setting. Here, we propose a
general method to construct explicit trajectories which explore the space state
with positive probability and witch permit to prove a petite-set condition for
the compact sets of the state space. An application to an age-structured
growthfragmentation process modelling bacterial growth is also shown.
