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Invariant Gibbs measure for Anderson nonlinear wave equation

Nikolay Barashkov, Francesco C. De Vecchi, Immanuel Zachhuber

TL;DR

The paper develops a rigorous framework for invariant Gibbs measures for the nonlinear wave equation with a random Anderson potential on $\mathbb{T}^2$. Central to the approach is a regular coupling between the Gaussian free field and the Anderson Gaussian free field, enabled by a Boué–Dupuis variational representation, which allows well-defined Wick powers and the construction of an Anderson $\Phi_2^4$ Gibbs measure. The authors establish local well-posedness for the Wick-ordered NLW, then globalize the dynamics via finite-dimensional approximations and a smoothing removal procedure, ultimately proving invariance of the Gibbs measure under the nonlinear flow. The results connect renormalization, paracontrolled calculus, and variational methods to yield a robust invariant measure theory for singular stochastic PDEs with random potentials, with potential implications for the continuum limits of disordered systems and related stochastic wave phenomena.

Abstract

We study the Gaussian measure whose covariance is related to the Anderson Hamiltonian operator, proving that it admits a regular coupling to the (standard) Gaussian free field exploiting the stochastic optimal control formulation of Gibbs measures. Using this coupling, we define the renormalized powers of the Anderson free field and we prove that the associated quartic Gibbs measure is invariant under the flow of a nonlinear wave equation with renormalized cubic nonlinearity.

Invariant Gibbs measure for Anderson nonlinear wave equation

TL;DR

The paper develops a rigorous framework for invariant Gibbs measures for the nonlinear wave equation with a random Anderson potential on . Central to the approach is a regular coupling between the Gaussian free field and the Anderson Gaussian free field, enabled by a Boué–Dupuis variational representation, which allows well-defined Wick powers and the construction of an Anderson Gibbs measure. The authors establish local well-posedness for the Wick-ordered NLW, then globalize the dynamics via finite-dimensional approximations and a smoothing removal procedure, ultimately proving invariance of the Gibbs measure under the nonlinear flow. The results connect renormalization, paracontrolled calculus, and variational methods to yield a robust invariant measure theory for singular stochastic PDEs with random potentials, with potential implications for the continuum limits of disordered systems and related stochastic wave phenomena.

Abstract

We study the Gaussian measure whose covariance is related to the Anderson Hamiltonian operator, proving that it admits a regular coupling to the (standard) Gaussian free field exploiting the stochastic optimal control formulation of Gibbs measures. Using this coupling, we define the renormalized powers of the Anderson free field and we prove that the associated quartic Gibbs measure is invariant under the flow of a nonlinear wave equation with renormalized cubic nonlinearity.
Paper Structure (18 sections, 61 theorems, 359 equations)

This paper contains 18 sections, 61 theorems, 359 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Anderson Hamiltonian and Paracontrolled Calculus
  4. Gaussian measures and Wick powers
  5. A (regular) coupling between GFF and AGFF
  6. Variational formulation of the coupling problem
  7. Some stochastic estimates
  8. Analytical estimates
  9. Bounds on $F^{\varepsilon}$
  10. Construction of the coupling
  11. On the renormalization of the powers of AGFF
  12. Local-in-time solution
  13. Local-in-time convergence of approximations
  14. Globalization and Invariance
  15. Approximation by finite dimensional system
  16. ...and 3 more sections

Key Result

Proposition 1

Let $0 \leqslant K (\omega) < + \infty$ be such that $\mathbb{H}^{\omega, K} =\mathbb{H}^{\omega} + K (\omega) + 1$ is a (strictly) positive operator and let $\mu^{\mathbb{H}^{\omega, K}}$ be the law of the Gaussian field with covariance $(\mathbb{H}^{\omega, K})^{- 1}$. Then, for almost every $\ome where $\mathcal{Z}_{\varepsilon}$ is the normalization constant, exists in $\mathcal{P} (H^{- \kapp

Theorems & Definitions (83)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Definition 1: and Lemma
  • Definition 2
  • Lemma 1
  • Theorem 3: Self-adjointness and (Form-)Domain of the Anderson Hamiltonian
  • Proposition 2: Norm resolvent convergence of approximate operators
  • Proposition 3
  • ...and 73 more