多维年龄依赖分支过程的长期行为：具有奇异跳跃核建模端粒缩短
Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel modelling telomere shortening
AI总结：
本文研究了具有奇异跳跃核的多维年龄依赖分支过程的长期行为，通过随机比较和弱Harnack不等式方法，证明了指数平稳性，并验证了主要结论的假设。
摘要：
在本文中，我们研究了一种多维年龄依赖分支过程的平稳行为，该过程受细胞群体中端粒缩短现象的启发。我们的模型追踪个体在连续时间框架中演变，由二叉树索引，特征为年龄和多维特征。分支事件的发生率依赖于年龄，其中子代继承特征时在某些坐标上随机增加或减少，而大部分则保持不变。尽管存在无界年龄依赖出生率（可能依赖于多维特征）和非紧致转移核，我们仍通过随机比较模型与Bellman-Harris过程，以及使用弱形式的Harnack不等式，获得了指数平稳性。本文最后通过例子验证了主要结果的假设。
英文摘要：
In this article, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified.
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