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On the negativity of the top Lyapunov exponent for stochastic differential equations driven by fractional Brownian motion

Alexandra Blessing Neamţu, Mazyar Ghani Varzaneh

TL;DR

The paper addresses the sign of the top Lyapunov exponent for stochastic differential equations driven by fractional Brownian motion, a non-Markovian setting where standard Fokker–Planck tools fail. It builds a robust SDS–RDS framework by coupling the SDE to a stationary fbm noise process and performing a disintegration-based construction of an invariant measure for the associated RDS. A key contribution is establishing Gaussian-type density bounds for the invariant measure under a rescaling that isolates the diffusion strength, enabling a Birkhoff ergodic-theorem argument to show the top Lyapunov exponent becomes negative for large noise intensity. This yields a non-Markovian analogue of stabilization by noise, with implications for stable manifolds, attraction, and potential synchronization phenomena in fbm-driven systems. The methods blend stochastic and random dynamical systems theory with fractional calculus and Wiener–Liouville bridge techniques to obtain quantitative control over the invariant density and Lyapunov spectra in a non-Markovian regime.

Abstract

We provide sign information for the top Lyapunov exponent for a stochastic differential equation driven by fractional Brownian motion.~To this aim we analyze the stochastic dynamical system generated by such an equation, obtain a random dynamical system and construct an appropriate invariant measure.~Suitable estimates for its density together with Birkhoff's ergodic theorem imply the negativity of the top Lyapunov exponent by increasing the noise intensity.

On the negativity of the top Lyapunov exponent for stochastic differential equations driven by fractional Brownian motion

TL;DR

The paper addresses the sign of the top Lyapunov exponent for stochastic differential equations driven by fractional Brownian motion, a non-Markovian setting where standard Fokker–Planck tools fail. It builds a robust SDS–RDS framework by coupling the SDE to a stationary fbm noise process and performing a disintegration-based construction of an invariant measure for the associated RDS. A key contribution is establishing Gaussian-type density bounds for the invariant measure under a rescaling that isolates the diffusion strength, enabling a Birkhoff ergodic-theorem argument to show the top Lyapunov exponent becomes negative for large noise intensity. This yields a non-Markovian analogue of stabilization by noise, with implications for stable manifolds, attraction, and potential synchronization phenomena in fbm-driven systems. The methods blend stochastic and random dynamical systems theory with fractional calculus and Wiener–Liouville bridge techniques to obtain quantitative control over the invariant density and Lyapunov spectra in a non-Markovian regime.

Abstract

We provide sign information for the top Lyapunov exponent for a stochastic differential equation driven by fractional Brownian motion.~To this aim we analyze the stochastic dynamical system generated by such an equation, obtain a random dynamical system and construct an appropriate invariant measure.~Suitable estimates for its density together with Birkhoff's ergodic theorem imply the negativity of the top Lyapunov exponent by increasing the noise intensity.

Paper Structure

This paper contains 18 sections, 50 theorems, 276 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Stochastic and random dynamical systems
  5. Fractional Brownian motion
  6. Stationary noise process associated to fractional Brownian motion
  7. Construction of a random dynamical system given a stochastic dynamical system
  8. Invariant measure for the RDS.
  9. Stochastic differential equations driven by fractional Brownian motion
  10. Well-posedness and the generation of a stochastic dynamical system
  11. Invariant measure for the SDS.
  12. Two approaches for the generation of RDS for SDEs with additive fractional noise
  13. A local stable manifold theorem
  14. Lyapunov exponents
  15. A rescaling argument
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $(X, \sigma(X))$ and $(Y, \sigma(Y))$ be two measure spaces, $T: X \to Y$ be a measurable function and let $\mu$ be a Borel measure on $X$. Then the pushforward measure $T_{\star} \mu$ on $Y$ is defined by In addition, for a measurable function $f: Y \to \mathbb{R}^+$, we have

Figures (1)

  • Figure 1: Visualization of $\vartheta_{t}\omega^+$ and $P_t(\omega^-, \omega^+)$ for two paths $\omega^-$ and $\omega^+$ in $\mathcal{B}_{H}$.

Theorems & Definitions (91)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 81 more