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Persistence of stability for equilibria of map iterations in Banach spaces under small random perturbations. (English) Zbl 1317.37011

From the text: Consider a closed subset \(S\) of a separable Banach space \((X,\| \cdot \|)\) and a map \(\phi : S\to S\) with suitable regularity [\(\dots{}\)]. Iteration of \(\phi\) yields a dynamical system in \(S\) in discrete time. It is randomly perturbed in the following way. Assume that, for each \(x\in S\), a Borel probability measure \(Q_x\) on \(X\) is given, which is the law of a random additive perturbation \(Y\) that can take place at state \(x\). We consider the long-term behavior of the Markov chain defined (informally) by \[ X_{n+1}=\phi (X_n)+Y_n,\quad Y_n\sim Q_{\phi (X_n)}, \] when \(X_0\) is taken in a neighbourhood of a stable fixed point \(x^*\) of \(\phi\).
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.

MSC:

37A50 Dynamical systems and their relations with probability theory and stochastic processes
47D07 Markov semigroups and applications to diffusion processes
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
92D25 Population dynamics (general)
60G51 Processes with independent increments; Lévy processes
34F05 Ordinary differential equations and systems with randomness
37N25 Dynamical systems in biology
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