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Synchronization of stochastic dissipative differential equation driven by fractional Brownian motions

Qiyong Cao, Hongjun Gao, Wei Wei

TL;DR

This work addresses synchronization of dissipative stochastic differential equations driven by nonlinear multiplicative fractional Brownian noise, with Hurst index $H \\in \\left(\\tfrac{1}{3},\\tfrac{1}{2}\\right)\\cup\\left(\\tfrac{1}{2},1\\right)$. It uses the Doss-Sussmann transformation to convert the stochastic system into a random differential equation framework, enabling trajectory synchronization for both Young ($H>\\tfrac{1}{2}$) and rough-path ($H\\in(\\tfrac{1}{3},\\tfrac{1}{2}]$) regimes under dissipativity and smoothness assumptions on the drift and diffusion coefficients. The main results show that, as the coupling strength $\\kappa$ tends to infinity, the two component trajectories converge to a common synchronized path $\\bar{Y}$, governed by the averaged drift $\\frac{1}{2}(f(\\bar{Y})+g(\\bar{Y}))$ with the same noise drive, in both the Young and rough settings. An illustrative double-well example validates the theory and demonstrates robustness under both standard fractional Brownian motion and rough-path formulations. This work extends synchronization mechanisms to nonlinear multiplicative fractional noise, providing a pathwise framework that complements existing Lyapunov and random-dynamical approaches.

Abstract

In this paper, we study a class of dissipative stochastic differential equations driven by nonlinear multiplicative fractional Brownian noise with Hurst index $H \in \left(\frac{1}{3},\frac{1}{2})\cup(\frac{1}{2}, 1\right) $. We establish the well-posedness of the associated coupled stochastic differential equations and prove synchronization in the sense of trajectories. Our approach relies on the Doss-Sussmann transformation, which enables us to extend existing results for additive and linear noise to the case of nonlinear multiplicative fractional Brownian noise. The findings provide new insights into the synchronization of dissipative systems under fractional noise perturbations.

Synchronization of stochastic dissipative differential equation driven by fractional Brownian motions

TL;DR

This work addresses synchronization of dissipative stochastic differential equations driven by nonlinear multiplicative fractional Brownian noise, with Hurst index . It uses the Doss-Sussmann transformation to convert the stochastic system into a random differential equation framework, enabling trajectory synchronization for both Young () and rough-path () regimes under dissipativity and smoothness assumptions on the drift and diffusion coefficients. The main results show that, as the coupling strength tends to infinity, the two component trajectories converge to a common synchronized path , governed by the averaged drift with the same noise drive, in both the Young and rough settings. An illustrative double-well example validates the theory and demonstrates robustness under both standard fractional Brownian motion and rough-path formulations. This work extends synchronization mechanisms to nonlinear multiplicative fractional noise, providing a pathwise framework that complements existing Lyapunov and random-dynamical approaches.

Abstract

In this paper, we study a class of dissipative stochastic differential equations driven by nonlinear multiplicative fractional Brownian noise with Hurst index . We establish the well-posedness of the associated coupled stochastic differential equations and prove synchronization in the sense of trajectories. Our approach relies on the Doss-Sussmann transformation, which enables us to extend existing results for additive and linear noise to the case of nonlinear multiplicative fractional Brownian noise. The findings provide new insights into the synchronization of dissipative systems under fractional noise perturbations.

Paper Structure

This paper contains 9 sections, 17 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $p$--variation space and Hölder space
  4. Young integral
  5. Rough integral
  6. Young and rough differential equations
  7. Synchronization
  8. Example
  9. The proof of Theorem \ref{['Sec:thm 3.1']}

Key Result

Lemma 2.1

Let $p\geq 1$ and $x\in C^{p-var}(I;\mathbb{R}^d)$. For any partition of $I$ given by $a=\tau_0<\tau_1<\cdots<\tau_{N}=b$, we have

Figures (2)

  • Figure 1: Trajectories of $Y^1_t$,$Y^2_t$,$Y_t$ for $\kappa=0,10,100,1000$.
  • Figure 2: Trajectories of $Y^1_t$,$Y^2_t$,$Y_t$ for $\kappa=0,10,100,1000$.

Theorems & Definitions (37)

  • Lemma 2.1: Lemma 2.1 MR3871629
  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.2
  • Remark 2.3
  • Definition 2.3
  • Theorem 2.1: MR4174393
  • Remark 2.4
  • Remark 3.1
  • ...and 27 more