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Synchronization by noise for stochastic differential equations driven by fractional Brownian motion

Alexandra Blessing, Mazyar Ghani Varzaneh

Abstract

We investigate synchronization by noise for stochastic differential equations (SDEs) driven by a fractional Brownian motion (fbm) with Hurst index $H\in(0,1)$. Provided that the SDE has a negative top Lyapunov exponent, we show that a weak form of synchronization occurs. To this aim we use tools from stochastic dynamical systems, random dynamical systems and a support theorem for SDEs driven by fractional noise.~In particular, we characterize the support of an invariant measure of a random dynamical system in a non-Markovian setting.

Synchronization by noise for stochastic differential equations driven by fractional Brownian motion

Abstract

We investigate synchronization by noise for stochastic differential equations (SDEs) driven by a fractional Brownian motion (fbm) with Hurst index . Provided that the SDE has a negative top Lyapunov exponent, we show that a weak form of synchronization occurs. To this aim we use tools from stochastic dynamical systems, random dynamical systems and a support theorem for SDEs driven by fractional noise.~In particular, we characterize the support of an invariant measure of a random dynamical system in a non-Markovian setting.
Paper Structure (18 sections, 28 theorems, 186 equations)

This paper contains 18 sections, 28 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Literature
  4. Structure of the paper
  5. Preliminaries
  6. Notations
  7. Stochastic and random dynamical systems
  8. Pullback attractors
  9. Fractional Brownian motion
  10. SDEs driven by fractional Brownian motion
  11. Random invariant measures and attractors for RDS
  12. Random invariant measures
  13. Random attractors
  14. Synchronization by noise
  15. Support of the invariant measure of the RDS
  16. ...and 3 more sections

Key Result

Theorem 1.2

The SDE sde:intro exhibits weak synchronization in the sense of Definition def:intro provided that $\sigma$ is large enough and $F$ satisfies Assumption AASQw98a.

Theorems & Definitions (62)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 52 more