论文标题
年龄大小的分段确定性过程的指数性千古性
Exponential ergodicity of a degenerate age-size piecewise deterministic process
论文作者
论文摘要
我们研究了非保守分段确定性度量值的长期行为，并通过随机跳跃时间之间的确定性流动驱动的R 2 +驱动，其过渡内核具有退化形式，并支持R 2 +。使用DOOB H变换，该功能H被视为相关发生器的特征功能，我们可以将自己带回对保守过程的研究，该过程通过Harris的定理证明了指数性的性格。特别关注海丁林的缩小条件的证明。主要的困难是两个变量之一的退化，以及两个变量之间的确定性依赖性，这使得在二维空间中相对于非偏差度量的轨迹的期望值并不是微不足道的，这在非划分设置中尤其困难。在这里，我们提出了一种构建显式轨迹的通用方法，该方法以正概率和巫婆允许探索空间状态，以证明状态空间的紧凑型集合。还显示了对年龄结构化生长抛弃过程建模细菌生长的应用。
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.
