This paper addresses the long-term behaviour -in a suitable probabilistic senseof map iteration in subsets of Banach spaces that are randomly perturbed. The law of the latter change in state is allowed to depend on state. We provide quite general conditions under which a stable fixed point of the deterministic map iteration induces an asymptotically stable ergodic measure of the Markov chain defined by the perturbed system, which is regarded as 'persistence of stability'. The support of this invariant measure is characterized. The applicability of the framework is illustrated for deterministic dynamical systems that are subject to random interventions at fixed equidistant time points. In particular, we consider systems motivated by population dynamics: a model in ordinary differential equations, a model derived from a reaction-diffusion system and a class of delay equations.Keywords Stochastic interventions · Deterministic dynamical system · Stability of Invariant Measure · Markov operator Mathematics Subject Classifications (2010) Primary 47D07 · 60J20 · 60J35 · 60J75 · Secondary 92D25 · 34F05 · 60G51
This paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on càdlàg functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.2010 Mathematics subject classification: primary 54E35; secondary 54C35.
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