Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singular stochastic delay equations driven by fractional Brownian motion: Dynamics, longtime behaviour, and pathwise stability

Mazyar Ghani Varzaneh, Sebastian Riedel

Abstract

We study differential equations with a linear, path dependent drift and discrete delay in the diffusion term driven by a $γ$-Hölder rough path for $γ> \frac{1}{3}$. We prove well-posedness of these systems and establish a priori bounds for their solutions. Applying these results to an equation driven by a multidimensional fractional Brownian motion with Hurst paramter $H \in \big(\frac{1}{3},1\big)$, we can prove the existence of a Lyapunov spectrum for the linearized system that describes its long-time behaviour. Furthermore, we can deduce the existence of local stable, unstable, and center manifolds for the nonlinear equation. As an application, we can prove that under suitable conditions, the solution exhibits pathwise local exponential stability. En passant, we present new, concise and relatively short proofs for some classical results for $\mathcal{C}_0$-semigroups that build on the Multiplicative Ergodic Theorem formulated on Banach spaces.

Singular stochastic delay equations driven by fractional Brownian motion: Dynamics, longtime behaviour, and pathwise stability

Abstract

We study differential equations with a linear, path dependent drift and discrete delay in the diffusion term driven by a -Hölder rough path for . We prove well-posedness of these systems and establish a priori bounds for their solutions. Applying these results to an equation driven by a multidimensional fractional Brownian motion with Hurst paramter , we can prove the existence of a Lyapunov spectrum for the linearized system that describes its long-time behaviour. Furthermore, we can deduce the existence of local stable, unstable, and center manifolds for the nonlinear equation. As an application, we can prove that under suitable conditions, the solution exhibits pathwise local exponential stability. En passant, we present new, concise and relatively short proofs for some classical results for -semigroups that build on the Multiplicative Ergodic Theorem formulated on Banach spaces.

Paper Structure

This paper contains 9 sections, 27 theorems, 221 equations.

Table of Contents

  1. Preliminaries
  2. Basic properties of rough delay equations
  3. Delayed Lévy Area for Fractional Brownian Motion with Smooth Approximations
  4. Existence, uniqueness, and some a priori bounds
  5. Random dynamical systems and invariant manifolds
  6. Pathwise stability results
  7. The nonlinear case
  8. Elements of Malliavin Calculus and fractional calculus
  9. Stochastic integrals

Key Result

Theorem 1.5

Assume $\mathbf{X} = (X, \mathbb{X}, \mathbb{X}(-r))$ is a delayed $\gamma$-rough path, and let $\frac{1}{3} < \beta \leq \gamma$ such that $2\beta + \gamma > 1$. Further assume that $\zeta$ is a $\beta$-Hölder, $L(\mathbb{R}^d, U)$-valued delayed controlled path based on $\mathbf{X}$, with the deco exists, where $\Pi$ is a partition of $[a, b]$. Moreover, there exists a constant $C$, depending on

Theorems & Definitions (83)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 73 more