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

Abstract

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

The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locations

“…Since ḡ is bounded, the above estimation also holds (with some Ĉ in the place of C) for n ≤ n 0 . We finally get ∞ n=1 E x 1 ,x 2 Z (1) n (ḡ) − Z (2) n (ḡ) < ∞ for every x 1 , x 2 ∈ X, which proves (3.23).…”

“…They are encountered as suitable mathematical models for processes in the physical world around us, e.g. in biology, as stochastic model for gene expression [25], gene regulation [18], excitable membranes [29] or population dynamics [1,2], as well as in resource allocation and service provisioning (queing, cf. [12]).…”

“…The case of non-locally compact state space has been studied much less so far (see e.g. [2,8,18,29,31]). A similar statement applies to the study of limit theorems (see [32,29]).…”

“…Here, and e.g. also in [2], only the jump is distributed conditionally given the current state of the system. The process examined in this paper (described in detail in Section 2) involves jumps that occure at random time points according to a Poisson process.…”

Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process

“…in resource allocation and service provisioning (queing, cf. [9]) or biology: as stochastic models for gene expression [20], cell division [19], gene regulation [15], excitable membranes [21] or population dynamics [1]. Mathematical research on PDMPs has been conducted over the years in various directions.…”