The Markov evolution is studied of an infinite age-structured population of migrants arriving in and departing from a continuous habitat $X\subseteq {ℝ}^d$—at random and independently of each other. Each population member is characterized by its age a ≥ 0 (time of presence in the population) and location x ∈ X. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is an explicit construction of the evolution μ₀ → μ_t of such states by solving a standard Fokker–Planck equation for this models. We also found a stationary state μ existing if the emigration rate is separated away from zero. Under additional conditions imposed on μ₀, it is shown that μ_t weakly converges to μ as t → +∞.

中文翻译：

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无限年龄结构粒子系统的动力学

马尔可夫演化研究了无限年龄结构的移民人口进入和离开连续栖息地$X\subseteq {ℝ}^d$——随机且相互独立。每个种群成员的特征在于其年龄a  ≥ 0（在种群中的存在时间）和位置x  ∈  X。总体状态是相应标记配置空间上的概率度量。本文的结果是通过求解该模型的标准 Fokker-Planck 方程，明确构造了这些状态的演化μ ₀  →  μ _{t 。}我们还发现，如果迁移率远离零，则存在静止状态μ 。在施加于μ ₀的附加条件下，表明μ _t当t  → + ∞时，弱收敛到μ。