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Anderson stochastic quantization equation

Hugo Eulry, Antoine Mouzard, Tristan Robert

Abstract

We study the parabolic defocusing stochastic quantization equation with both mutliplicative spatial white noise and an independant space-time white noise forcing, on compact surfaces, with polynomial nonlinearity. After renormalizing the nonlinearity, we construct the random Gibbs measure as an absolutely continuous measure with respect to the law of the Anderson Gaussian Free Field for fixed realization of the spatial white noise. Then, when the initial data is distributed according to the Gibbs measure, we prove almost sure global well-posedness for the dynamics and invariance of the Gibbs measure.

Anderson stochastic quantization equation

Abstract

We study the parabolic defocusing stochastic quantization equation with both mutliplicative spatial white noise and an independant space-time white noise forcing, on compact surfaces, with polynomial nonlinearity. After renormalizing the nonlinearity, we construct the random Gibbs measure as an absolutely continuous measure with respect to the law of the Anderson Gaussian Free Field for fixed realization of the spatial white noise. Then, when the initial data is distributed according to the Gibbs measure, we prove almost sure global well-posedness for the dynamics and invariance of the Gibbs measure.
Paper Structure (18 sections, 30 theorems, 269 equations)

This paper contains 18 sections, 30 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. A singular SPDE in a singular random environment
  3. Main results and sketch of the proof
  4. Construction of the stochastic objects
  5. Algebraic setting and Wick powers
  6. Renormalization
  7. Truncated Gibbs measure and convergence
  8. Well-posedness theory
  9. Construction of the dynamics
  10. Deterministic local well-posedness
  11. Almost sure globalization and measure invariance
  12. A singular case: the Anderson Hamiltonian
  13. Construction of the operator
  14. Anderson Green's function and GFF
  15. Comparison of semi-classical multipliers
  16. ...and 3 more sections

Key Result

Theorem 1.1

The sequence $(\rho_N)_N$ converges in total variation to a measure $\rho$ that is absolutely continuous with respect to $\mu^\mathcal{H}$.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 49 more