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.

我们研究了非保守分段确定性测值马尔可夫过程的长期行为，该过程模拟了年龄和规模结构群体的增殖，概括了细菌生长的“加法器”模型。首先，我们证明相关无穷小生成器的特征元素的存在性，这用于使我们回到使用 Doob \(h\)来研究保守马尔可夫过程-转换。最后，我们通过漂移小化论证获得了该过程的指数遍历性。具体来说，我们展示了状态空间紧集的“娇小性”。这允许克服当尝试在仅具有平流和简并跳跃项的无界二维空间上在固定均匀时间构建混合轨迹时遇到的困难。