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A mild rough Gronwall Lemma with applications to non-autonomous evolution equations

Alexandra Blessing, Mazyar Ghani Varzaneh, Tim Seitz

TL;DR

This work develops a mild Gronwall-type inequality for mild solutions of non-autonomous parabolic RPDEs driven by Gaussian Volterra rough paths, enabling integrable moment bounds for both the solution and its Jacobian. By refining the sewing lemma with control functions and analyzing the Cameron-Martin space of the noise, the authors derive the moment bounds needed to apply the multiplicative ergodic theorem and obtain Lyapunov exponents for RPDEs. They prove independence of these exponents from the interpolation-space norm and establish invariant structures such as stable manifolds via a random dynamical system framework in infinite dimensions. The results are illustrated with RPDEs having time-dependent coefficients and multiplicative rough boundary noise, highlighting the practical impact for ergodicity, stability, and long-time behavior in stochastic PDEs with rough inputs.

Abstract

We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to obtain the existence of moments of all order for the solution of the corresponding RPDE and its Jacobian when the random input is given by a Gaussian Volterra process. Applying further the multiplicative ergodic theorem, these integrable bounds entail the existence of Lyapunov exponents for RPDEs. We illustrate these results for stochastic partial differential equations with multiplicative boundary noise.

A mild rough Gronwall Lemma with applications to non-autonomous evolution equations

TL;DR

This work develops a mild Gronwall-type inequality for mild solutions of non-autonomous parabolic RPDEs driven by Gaussian Volterra rough paths, enabling integrable moment bounds for both the solution and its Jacobian. By refining the sewing lemma with control functions and analyzing the Cameron-Martin space of the noise, the authors derive the moment bounds needed to apply the multiplicative ergodic theorem and obtain Lyapunov exponents for RPDEs. They prove independence of these exponents from the interpolation-space norm and establish invariant structures such as stable manifolds via a random dynamical system framework in infinite dimensions. The results are illustrated with RPDEs having time-dependent coefficients and multiplicative rough boundary noise, highlighting the practical impact for ergodicity, stability, and long-time behavior in stochastic PDEs with rough inputs.

Abstract

We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to obtain the existence of moments of all order for the solution of the corresponding RPDE and its Jacobian when the random input is given by a Gaussian Volterra process. Applying further the multiplicative ergodic theorem, these integrable bounds entail the existence of Lyapunov exponents for RPDEs. We illustrate these results for stochastic partial differential equations with multiplicative boundary noise.

Paper Structure

This paper contains 20 sections, 40 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Preliminaries. Rough path theory and parabolic evolution families
  3. Existence and integrable bounds of global solutions
  4. Sewing lemma revisited
  5. An integrable a-priori bound
  6. Cameron-Martin space associated to the noise
  7. Rough Gronwall's inequality
  8. The mild Gronwall Lemma
  9. Linearization of the rough PDE
  10. An application. Lyapunov exponents for random dynamical systems
  11. Generation of a random dynamical system
  12. Multiplicative ergodic theorem
  13. Independence of the Lyapunov exponents on the norm of the interpolation spaces
  14. Invariant manifolds
  15. Examples
  16. ...and 5 more sections

Key Result

Theorem 2.4

(AT87) Let $(A(t))_{t\in [0,T]}$ satisfy Assumption ass:A1-ass:A3. Then there exists a unique parabolic evolution family $(U_{t,s})_{0\leq s\leq t\leq T}$ of linear operators $U_{t,s}:E_0\to E_0$ such that the following properties hold:

Theorems & Definitions (111)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 101 more