Stochastic functional Kolmogorov equations, I: Persistence
Stochastic functional Kolmogorov equations, I: Persistence
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DOI：
10.1016/j.spa.2021.09.007
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发表时间：
2021-05
影响因子：
1.4
通讯作者：
D. Nguyen;N. Nguyen;G. Yin
中科院分区：
文献类型：
--
作者：
D. Nguyen;N. Nguyen;G. Yin
这项工作（第一部分）连同其同伴（第二部分）开发了一个新的框架随机泛函Kolmogorov方程，这是非线性随机微分方程依赖于当前以及过去的状态。由于结果的复杂性，将我们的贡献分为两部分似乎是有益的。与现有文献相比，我们的努力是通过允许延迟和过去的依赖性来推进知识，从而为广泛的应用提供必要的实用性。生物学和生态学中一个长期存在的基本问题是：一个种群长期存在和灭绝（或相互作用的物种长期共存）的最小必要和充分条件是什么？不管遇到的具体应用，持久性和灭绝是共同的Kolmogorov系统的属性。虽然基于随机微分方程的Kolmogorov方程已经有了许多优秀的研究成果，但关于具有过去依赖性的随机Kolmogorov方程的研究工作仍然很少。我们的目标是回答上述基本问题。本书的第一部分致力于描述持久性，而它的同伴第二部分则致力于灭绝。在本文中使用的主要技术包括新开发的功能伊藤公式和渐近耦合和Harris型理论的无穷维系统专门功能方程。首先建立了随机泛函Kolmogorov方程的一般定理。然后，一些应用程序进行检查，覆盖，改进，并大大扩展现有的文献。此外，我们的结果减少到现有文献中的Kolmogorov系统时，没有过去的依赖。
This work (Part (I)) together with its companion (Part (II)) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the results, it seems to be instructive to divide our contributions to two parts. In contrast to the existing literature, our effort is to advance the knowledge by allowing delay and past dependence, yielding essential utility to a wide range of applications. A long-standing question of fundamental importance pertaining to biology and ecology is: What are the minimal necessary and sufficient conditions for long-term persistence and extinction (or for long-term coexistence of interacting species) of a population? Regardless of the particular applications encountered, persistence and extinction are properties shared by Kolmogorov systems. While there are many excellent treaties of stochastic-differential-equation-based Kolmogorov equations, the work on stochastic Kolmogorov equations with past dependence is still scarce. Our aim here is to answer the aforementioned basic question. This work, Part (I), is devoted to characterization of persistence, whereas its companion, Part (II) is devoted to extinction. The main techniques used in this paper include the newly developed functional Itô formula and asymptotic coupling and Harris-like theory for infinite dimensional systems specialized to functional equations. General theorems for stochastic functional Kolmogorov equations are developed first. Then a number of applications are examined covering, improving, and substantially extending the existing literature. Furthermore, our results reduce to that in the existing literature of Kolmogorov systems when there is no past dependence.