We provide quantitative estimates in total variation distance for positive semigroups, which can be non-conservative and non-homogeneous. The techniques relies on a family of conservative semigroups that describes a typical particle and Doeblin’s type conditions inherited from Champagnat and Villemonais (Probab. Theory Relat. Fields 164(1–2):243–283, 2016) for coupling the associated process. Our aim is to provide quantitative estimates for linear partial differential equations and we develop several applications for population dynamics in varying environment. We start with the asymptotic profile for a growth diffusion model with time and space non-homogeneity. Moreover we provide general estimates for semigroups which become asymptotically homogeneous, which are applied to an age-structured population model. Finally, we obtain a speed of convergence for periodic semigroups and new bounds in the homogeneous setting. They are illustrated on the renewal equation.

中文翻译：

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广义Doeblin条件下非保守半群的遍历行为

我们为正半群提供了总变化距离的定量估计，这可以是非保守和非均质的。该技术依赖于一个保守的半群族，该族描述了典型的粒子和Doeblin的类型条件（继承自Champagnat和Villemonais）（Probab。Theory Relat。Fields 164（1-2）：243-283，2016），用于关联相关过程。我们的目的是为线性偏微分方程提供定量估计，并且我们为变化环境中的种群动态开发了几种应用程序。我们从具有时间和空间非均匀性的生长扩散模型的渐近轮廓开始。此外，我们提供了渐近同质的半群的一般估计，并将其应用于年龄结构的人口模型。最后，我们获得了周期半群和齐次布设中新边界的收敛速度。它们在续订方程式中说明。