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Caputo mean-square attractors for non-autonomous stochastic differential equations

Lijuan Zhang, Jianhua Huang, Yejuan Wang

TL;DR

The paper introduces Caputo mean-square attractors for non-autonomous stochastic evolution and provides a criterion for their existence. It develops a framework of Caputo mean-square random semi-dynamical systems and demonstrates existence results via uniform absorbing sets and asymptotic compactness. Through a detailed non-autonomous Caputo FSDE example driven by tempered fractional noise, the authors construct a Caputo mean-square random semi-dynamical system with a skew-product flow and prove the existence of a Caputo mean-square attractor in the weighted space $\mathfrak{C}_{\alpha} \times P$, along with a corresponding attractor for the cocycle. The results offer a rigorous foundation for the long-time behavior of non-autonomous, memory-bearing stochastic systems and provide tools applicable to fractional stochastic dynamics with non-Markovian noise.

Abstract

This paper investigates Caputo mean-square attractors for non-autonomous stochastic evolution systems. We first introduce the concept of Caputo mean-square attractors and then establish a sufficient criterion for existence of such attractors.As an application, we consider a non-autonomous Caputo fractional stochastic differential equation of order $α\in (\frac{1}{2},1)$ in $L^2(Ω; \mathbb{R}^d)$ with a driving system on a compact base space $P$ and tempered fractional noise.It is shown that this equation generates a Caputo mean-square random semi-dynamical system on $\mathfrak{C} \times P$ with a skew-product semi-flow structure,where $\mathfrak{C}$ denotes the space of continuous functions $f\in \mathbb{R}^{+}\rightarrow L^2(Ω; \mathbb{R}^d)$. Under suitable conditions, we prove that this semi-dynamical system admits a Caputo mean-square attractor.

Caputo mean-square attractors for non-autonomous stochastic differential equations

TL;DR

The paper introduces Caputo mean-square attractors for non-autonomous stochastic evolution and provides a criterion for their existence. It develops a framework of Caputo mean-square random semi-dynamical systems and demonstrates existence results via uniform absorbing sets and asymptotic compactness. Through a detailed non-autonomous Caputo FSDE example driven by tempered fractional noise, the authors construct a Caputo mean-square random semi-dynamical system with a skew-product flow and prove the existence of a Caputo mean-square attractor in the weighted space , along with a corresponding attractor for the cocycle. The results offer a rigorous foundation for the long-time behavior of non-autonomous, memory-bearing stochastic systems and provide tools applicable to fractional stochastic dynamics with non-Markovian noise.

Abstract

This paper investigates Caputo mean-square attractors for non-autonomous stochastic evolution systems. We first introduce the concept of Caputo mean-square attractors and then establish a sufficient criterion for existence of such attractors.As an application, we consider a non-autonomous Caputo fractional stochastic differential equation of order in with a driving system on a compact base space and tempered fractional noise.It is shown that this equation generates a Caputo mean-square random semi-dynamical system on with a skew-product semi-flow structure,where denotes the space of continuous functions . Under suitable conditions, we prove that this semi-dynamical system admits a Caputo mean-square attractor.
Paper Structure (10 sections, 9 theorems, 85 equations)

This paper contains 10 sections, 9 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Caputo mean-square random semi-dynamical system
  3. Example
  4. Existence of Attractors for Caputo FSDE \ref{['eq0.1']}
  5. Bounds on the solution
  6. A uniformly $\mathscr{D}$-absorbing set of $T_t$ in $\mathfrak{C}_\alpha$
  7. Uniform $\mathscr{D}$-asymptotic compactness of $T_t$ in $\mathfrak{C}_\alpha$
  8. Equi-Lipschitz continuity of solution
  9. Applying Ascoli's theorem
  10. Continuity of $\chi^*(\theta)$ at $\theta=0$

Key Result

Theorem 2.4

Let $\mathscr{D}$ be a collection of some bounded sets in $L^{2}(\Omega; \mathbb{R}^d)$. Suppose that the Caputo FSDE eq00.1 generates a Caputo MS-RSDS $\phi$ on $L^{2}(\Omega; \mathbb{R}^d)$. If $\phi$ is $\mathscr{D}$-asymptotically compact in $L^{2}(\Omega; \mathbb{R}^d)$ and has a bounded and cl Then, $\mathcal{A}$ is a closed and bounded $\mathscr{D}$-attractor of the Caputo MS-RSDS $\phi$, a

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 10 more