  This work (Part (I)) together with its companion (Part (II) [45]) 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)
[45], is devoted to extinction. The main techniques used in this paper include
the newly developed functional It^o 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 to obtain new
results substantially covering, improving, and extending the existing
literature. Furthermore, these conditions reduce to that of Kolmogorov systems
when there is no past dependence.
