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Averaging principle for slow-fast systems of PDEs with rough drivers

Miaomiao Li, Bin Pei, Yong Xu, Xiaole Yue

TL;DR

This work addresses averaging for slow–fast systems of rough PDEs driven by irregular signals on a monotone interpolation Hilbert-space scale. It develops a semigroup-free controlled rough path framework on the scale $(\mathcal{H}_\gamma)$ and employs a Khasminskii-type time discretization to compare the slow and averaged dynamics without relying on the underlying semigroup. The main result proves strong convergence in $L^2(\Omega; C([0,T]; \mathcal{H}_\gamma))$ of $X_t^\varepsilon$ to the averaged solution $\bar{X}_t$, with the averaged drift given by $\bar{F}_1(x)=\int F_1(x,y)\mu^x(dy)$ where $\mu^x$ is the invariant measure of the frozen fast equation. This extends averaging theory to rough PDEs with irregular drivers and relaxes spatial regularity requirements, enabling reduced models for multiscale systems in physics and engineering. The approach integrates rough-path techniques with ergodic averaging to provide a rigorous, pathwise, and quantitative convergence result.

Abstract

This paper investigates a class of slow--fast systems of rough partial differential equations defined over a monotone family of interpolation Hilbert spaces. By employing the controlled rough path framework tailored to a monotone family of interpolation spaces, together with a time discretization argument, we demonstrate that the slow component strongly converges to the solution of the averaged system in the supremum norm as the time-scale parameter $\varepsilon$ tends to $0$.

Averaging principle for slow-fast systems of PDEs with rough drivers

TL;DR

This work addresses averaging for slow–fast systems of rough PDEs driven by irregular signals on a monotone interpolation Hilbert-space scale. It develops a semigroup-free controlled rough path framework on the scale and employs a Khasminskii-type time discretization to compare the slow and averaged dynamics without relying on the underlying semigroup. The main result proves strong convergence in of to the averaged solution , with the averaged drift given by where is the invariant measure of the frozen fast equation. This extends averaging theory to rough PDEs with irregular drivers and relaxes spatial regularity requirements, enabling reduced models for multiscale systems in physics and engineering. The approach integrates rough-path techniques with ergodic averaging to provide a rigorous, pathwise, and quantitative convergence result.

Abstract

This paper investigates a class of slow--fast systems of rough partial differential equations defined over a monotone family of interpolation Hilbert spaces. By employing the controlled rough path framework tailored to a monotone family of interpolation spaces, together with a time discretization argument, we demonstrate that the slow component strongly converges to the solution of the averaged system in the supremum norm as the time-scale parameter tends to .
Paper Structure (13 sections, 17 theorems, 194 equations)

This paper contains 13 sections, 17 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Settings
  4. Rough path theory
  5. Main result
  6. Slow--fast system of rough PDEs
  7. Deterministic aspects
  8. Probabilistic aspects
  9. Frozen equation
  10. Proof of main result
  11. A time increment estimate and auxiliary process
  12. Proof of Theorem \ref{['MainRes']}
  13. Appendix

Key Result

Lemma 2.1

(GHN21). Let $\gamma \in \mathbf{R}$ and let $\mathbf{X}=(X,\mathbb{X}) \in \mathscr{C}^{\alpha}([0,T];\mathbf{R}^{d})$ for some $\alpha \in (1/3, 1/2]$. Let $(Y,Y^{\prime}) \in \mathcal{D}_{X}^{2\alpha}([0,T];\mathcal{H}_{\gamma}^d)$. Then the rough integral defined by exists in $\mathcal{H}_{\gamma}$, where the limit is taken along any partitions $\pi$ of $[0,t]$ and is independent of the speci

Theorems & Definitions (25)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.3
  • ...and 15 more