Stochastic functional Kolmogorov equations, I: Persistence

This work (Part (I)) together with its companion (II)) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear differential equations depending on the current as well past states. Because of complexity results, it seems to be instructive divide our contributions two parts. In contrast existing literature, effort is advance knowledge by allowing delay and dependence, yielding essential utility wide range applications. A long-standing question fundamental importance pertaining biology ecology is: What minimal necessary sufficient conditions long-term persistence extinction (or coexistence interacting species) population? Regardless particular applications encountered, properties shared systems. While there many excellent treaties stochastic-differential-equation-based dependence still scarce. Our aim here answer aforementioned basic question. work, Part (I), devoted characterization persistence, whereas companion, (II) extinction. The main techniques used in this paper include newly developed Itô formula asymptotic coupling Harris-like theory infinite dimensional systems specialized equations. General theorems first. Then number examined covering, improving, substantially extending literature. Furthermore, results reduce that literature when no dependence.