标题： 关于超临界一般分枝过程的强大数定律和Llog L条件 显示英文标题

标题： On the strong law of large numbers and Llog L condition for supercritical general branching processes

摘要： 我们考虑结构种群的分支过程：每个个体由一个属于一般可测状态空间的类型或特征来表征。 我们关注超临界遍历情况，其中种群可能存活并增长，且特征分布收敛。 然后，分支过程将由第一矩半群的第一特征值问题的正三元组驱动。 在重整化半群在加权总变差范数下收敛的假设下，我们证明了归一化经验测度的强收敛性和极限鞅的非退化性。 在Llog L条件下获得收敛性，该条件在无限维情况下提供了Kesten-Stigum结果，并放松了Asmussen和Hering在1976年工作中所需的重整化第一矩半群的均匀收敛假设。 证明的技术结合了马尔可夫族和半群的收缩以及Asmussen和Hering的截断过程。 我们还获得了重整化经验测度的L^1收敛性，并有助于统一文献中的不同结果。 这些结果大大扩展了大数定律适用的例子类别，我们通过吸收分支扩散、卡片屋模型和一些生长-分裂过程来说明这一点。 显示英文摘要

摘要： We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the normalized empirical measure and non-degeneracy of the limiting martingale. Convergence is obtained under an Llog L condition which provides a Kesten-Stigum result in infinite dimension and relaxes the uniform convergence assumption of the renormalized first moment semigroup required in the work of Asmussen and Hering in 1976. The techniques of proofs combine families of martingales and contraction of semigroups and the truncation procedure of Asmussen and Hering. We also obtain L^1 convergence of the renormalized empirical measure and contribute to unifying different results in the literature. These results greatly extend the class of examples where a law of large numbers applies, as we illustrate it with absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.