Exponential Ergodicity of a Degenerate Age-Size Piecewise Deterministic Process

We study the long-time behaviour of a non conservative piecewise deterministic measure-valued Markov process modelling the proliferation of an age-and-size structured population, which generalises the “adder” model of bacterial growth. Firstly, we prove the existence of eigenelements of the associated infinitesimal generator, which are used to bring ourselves back to the study of a conservative Markov process using a Doob \(h\)-transform. Finally, we obtain the exponential ergodicity of the process via drift-minorisation arguments. Specifically, we show the “petiteness” of the compact sets of the state space. This permits to circumvent the difficulties encountered when trying to construct mixing trajectories at a fixed uniform time on an unbounded two-dimensional space with only advection and degenerate jump terms.