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 proliferation an age-and-size structured population, which generalises “adder” model bacterial growth. Firstly, we prove existence eigenelements associated infinitesimal generator, are used to bring ourselves back using Doob $h$ -transform. Finally, obtain exponential ergodicity via drift-minorisation arguments. Specifically, show “petiteness” compact sets state space. This permits circumvent difficulties encountered when trying construct mixing trajectories at fixed uniform time on unbounded two-dimensional space with only advection and degenerate jump terms.