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$L^p$-Boltzmann-Gibbs principle via Littlewood-Paley-Stein inequality

Tadahisa Funaki

TL;DR

The paper develops an L^p Boltzmann-Gibbs principle for a one-dimensional asymmetric Ginzburg-Landau interface model by leveraging the Littlewood-Paley-Stein inequality. It provides second- and first-order replacement estimates with explicit, uniform-in-size error bounds, applicable to both strong and weak asymmetry regimes, under convexity assumptions on the potential. The methodology fuses Itô-Tanaka (Kipnis-Varadhan) arguments, a localized Sobolev-type LP-Stein framework, an L^p variational formula, and an L^p equivalence of ensembles to control fluctuations at multiple scales. These results contribute to a robust, high-integrability understanding of equilibrium fluctuations and are pertinent to KPZ-type weak universality analyses.

Abstract

In this paper, we establish the Boltzmann-Gibbs principle in the $L^p$ sense by applying the Littlewood-Paley-Stein inequality. Our model is an asymmetric Ginzburg-Landau interface model on a one-dimensional periodic lattice. Assuming convexity of the potential,we derive detailed error estimates, particularly their dependence on the size of the system and the size of the region on which the sample average is taken. Notably, the estimates are uniform in the strength of the asymmetry.

$L^p$-Boltzmann-Gibbs principle via Littlewood-Paley-Stein inequality

TL;DR

The paper develops an L^p Boltzmann-Gibbs principle for a one-dimensional asymmetric Ginzburg-Landau interface model by leveraging the Littlewood-Paley-Stein inequality. It provides second- and first-order replacement estimates with explicit, uniform-in-size error bounds, applicable to both strong and weak asymmetry regimes, under convexity assumptions on the potential. The methodology fuses Itô-Tanaka (Kipnis-Varadhan) arguments, a localized Sobolev-type LP-Stein framework, an L^p variational formula, and an L^p equivalence of ensembles to control fluctuations at multiple scales. These results contribute to a robust, high-integrability understanding of equilibrium fluctuations and are pertinent to KPZ-type weak universality analyses.

Abstract

In this paper, we establish the Boltzmann-Gibbs principle in the sense by applying the Littlewood-Paley-Stein inequality. Our model is an asymmetric Ginzburg-Landau interface model on a one-dimensional periodic lattice. Assuming convexity of the potential,we derive detailed error estimates, particularly their dependence on the size of the system and the size of the region on which the sample average is taken. Notably, the estimates are uniform in the strength of the asymmetry.

Paper Structure

This paper contains 24 sections, 18 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. One-dimensional Ginzburg-Landau interface model
  3. Height process $\phi_t$
  4. Tilt process $\eta_t$
  5. Stationarity, symmetric part and carré du champ
  6. The process $\phi_t$
  7. The process $\eta_t$
  8. Conservation law and canonical measures
  9. Boltzmann-Gibbs principle in the $L^p$ sense
  10. Main result
  11. Outline of the proof of Theorem \ref{['BG-2-B']}
  12. Several tools for the proof of Theorem \ref{['BG-2-B']}
  13. Itô-Tanaka trick in the $L^p$ sense in equilibrium
  14. Localization of symmetric $\eta$-process and its reversible measures
  15. The $\phi$-process
  16. ...and 9 more sections

Key Result

Lemma 2.1

For functions $F=F(\phi)$ and $G=G(\phi) \in C^2({\mathbb{R}}^N)$, if all integrals in the following converge, we have where $D_\phi F = (\partial_{\phi(x)} F)_{x\in {\mathbb{T}}_N}$.

Theorems & Definitions (43)

  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 33 more