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Admissibility and generalized nonuniform dichotomies for nonautonomous Random Dynamical Systems

Davor Dragicevic, Cesar M. Silva, Helder Vilarinho

Abstract

In this paper, we introduce generalized dichotomies for nonautonomous random linear dynamical systems acting on arbitrary Banach spaces, and obtain their complete characterization in terms of an appropriate admissibility property. These generalized dichotomies are associated to growth rates satisfying mild conditions and they include the standard exponential behavior as a very particular case. As a nontrivial application, we establish the robustness property of such dichotomies under small (linear) perturbations.

Admissibility and generalized nonuniform dichotomies for nonautonomous Random Dynamical Systems

Abstract

In this paper, we introduce generalized dichotomies for nonautonomous random linear dynamical systems acting on arbitrary Banach spaces, and obtain their complete characterization in terms of an appropriate admissibility property. These generalized dichotomies are associated to growth rates satisfying mild conditions and they include the standard exponential behavior as a very particular case. As a nontrivial application, we establish the robustness property of such dichotomies under small (linear) perturbations.

Paper Structure

This paper contains 7 sections, 13 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. NRDS and $\mu$-dichotomies
  3. Admissibility with respect to a random norm
  4. Technical lemma
  5. From dichotomies to admissibility and vice-versa
  6. Robustness
  7. Relation with random nonuniform $(\mu,\nu)$-dichotomy

Key Result

Lemma 3.1

Let $\mu$ be a growth rate. If $\alpha \in \ (0,1) \ \cup \ (1,+\infty)$, for each $r,s \in \mathbb{Z}^+$ with $r \ge s >1$, we have and, for $\alpha=1$, we have

Theorems & Definitions (34)

  • Remark 1
  • Remark 2
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 24 more