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Tiny fluctuations of the averaging process around its degenerate steady state

Federico Sau

TL;DR

This work analyzes nonequilibrium fluctuations of the degenerate averaging process on the discrete torus, revealing a nonstandard CLT scaling θ_ε=ε^{-(d/2+1)} that yields Gaussian fluctuations with explicit space-time correlations. The main technical innovations combine martingale methods with a LLN for weighted squared discrete gradients, proven via a Poincaré inequality in Poisson space and Malliavin calculus, to obtain variance bounds without higher-moment assumptions. The fluctuations converge to a Gaussian process solving the SPDE dY_t = 1/2 ΔY_t dt - 1/2 ∇·(ξ ∇u_t) dt, where u_t solves the heat equation and ξ is a matrix-valued space-time white noise with a non-diagonal covariance structure for d≥2. The paper thus bridges stochastic homogenization with nonequilibrium fluctuations in a degenerate setting, providing quantitative variance estimates and an explicit noise correlation that informs the understanding of dynamic interfaces and related averaging dynamics.

Abstract

We analyze nonequilibrium fluctuations of the averaging process on $\mathbb T_\varepsilon^d$, a continuous degenerate Gibbs sampler running over the edges of the discrete $d$-dimensional torus. We show that, if we start from a smooth deterministic non-flat interface, recenter, blow-up by a non-standard CLT-scaling factor $θ_\varepsilon=\varepsilon^{-(d/2+1)}$, and rescale diffusively, Gaussian fluctuations emerge in the limit $\varepsilon\to 0$. These fluctuations are purely dynamical, zero at times $t=0$ and $t=\infty$, and non-trivial for $t\in (0,\infty)$. We fully determine the correlation matrix of the limiting noise, non-diagonal as soon as $d\ge 2$. The main technical challenge in this stochastic homogenization procedure lies in a LLN for a weighted space-time average of squared discrete gradients. We accomplish this through a Poincaré inequality with respect to the underlying randomness of the edge updates, a tool from Malliavin calculus in Poisson space. This inequality, combined with sharp gradients' second moment estimates, yields quantitative variance bounds without prior knowledge of the limiting mean. Our method avoids higher (e.g., fourth) moment bounds, which seem inaccessible with the present techniques.

Tiny fluctuations of the averaging process around its degenerate steady state

TL;DR

This work analyzes nonequilibrium fluctuations of the degenerate averaging process on the discrete torus, revealing a nonstandard CLT scaling θ_ε=ε^{-(d/2+1)} that yields Gaussian fluctuations with explicit space-time correlations. The main technical innovations combine martingale methods with a LLN for weighted squared discrete gradients, proven via a Poincaré inequality in Poisson space and Malliavin calculus, to obtain variance bounds without higher-moment assumptions. The fluctuations converge to a Gaussian process solving the SPDE dY_t = 1/2 ΔY_t dt - 1/2 ∇·(ξ ∇u_t) dt, where u_t solves the heat equation and ξ is a matrix-valued space-time white noise with a non-diagonal covariance structure for d≥2. The paper thus bridges stochastic homogenization with nonequilibrium fluctuations in a degenerate setting, providing quantitative variance estimates and an explicit noise correlation that informs the understanding of dynamic interfaces and related averaging dynamics.

Abstract

We analyze nonequilibrium fluctuations of the averaging process on , a continuous degenerate Gibbs sampler running over the edges of the discrete -dimensional torus. We show that, if we start from a smooth deterministic non-flat interface, recenter, blow-up by a non-standard CLT-scaling factor , and rescale diffusively, Gaussian fluctuations emerge in the limit . These fluctuations are purely dynamical, zero at times and , and non-trivial for . We fully determine the correlation matrix of the limiting noise, non-diagonal as soon as . The main technical challenge in this stochastic homogenization procedure lies in a LLN for a weighted space-time average of squared discrete gradients. We accomplish this through a Poincaré inequality with respect to the underlying randomness of the edge updates, a tool from Malliavin calculus in Poisson space. This inequality, combined with sharp gradients' second moment estimates, yields quantitative variance bounds without prior knowledge of the limiting mean. Our method avoids higher (e.g., fourth) moment bounds, which seem inaccessible with the present techniques.
Paper Structure (24 sections, 23 theorems, 201 equations)

This paper contains 24 sections, 23 theorems, 201 equations.

Table of Contents

  1. Introduction, model and main results
  2. Averaging process
  3. Tiny nonequilibrium fluctuations
  4. A law of large numbers for the squared gradients
  5. Proof ideas and techniques
  6. Poincaré inequality in Poisson space
  7. Application of Poincaré inequality
  8. Ultracontractivity and gradient estimates for the averaging process
  9. Structure of the paper
  10. Discrete gradients
  11. Stochastic dynamics of discrete gradients
  12. Second moments of discrete gradients
  13. Technical estimates
  14. Proof of Theorem \ref{['th:LLN']}. Variance via Poincaré inequality
  15. Dynamics of the discrepancy
  16. ...and 9 more sections

Key Result

Theorem 1

Fix $\alpha>3+d/2$ and $u_0\in \mathcal{C}^2(\mathbb{T}^d)$. Then, there exists $\mathfrak a=\mathfrak a(d)\in (0,1]$ such that, when initializing the averaging process $(u_t^\varepsilon)_{t\ge 0}$ with $u_0^\varepsilon = u_0\rvert_{\mathbb{T}_\varepsilon^d}$, the following convergence in law holds true for the corresponding fluctuation fields $(\mathcal{Y}_t^\varepsilon)_{t\ge 0}$ given in eq:fl

Theorems & Definitions (50)

  • Theorem 1: Nonequilibrium fluctuations
  • Remark 1.1: Constant $\mathfrak a$
  • Theorem 2: LLN for squared discrete gradients
  • Lemma 2.1: Mild solution
  • proof
  • Lemma 2.2
  • Remark 2.1: Notation
  • Proposition 2.1
  • proof
  • Lemma 2.3
  • ...and 40 more