Persistence of Stability for Equilibria of Map Iterations in Banach Spaces Under Small Random Perturbations

This paper addresses the long–term behaviour –in a suitable probabilistic sense– of 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.