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Invariance of $φ^4$ measure under nonlinear wave and Schrödinger equations on the plane

Nikolay Barashkov, Petri Laarne

TL;DR

The work investigates invariance properties of the two-dimensional $\phi^4$ measure under nonlinear wave and Schrödinger dynamics on the plane. It combines stochastic quantization, Wick renormalization, and Besov-space techniques to transfer periodic invariance results to infinite volume via finite speed of propagation, yielding almost sure global well-posedness for the cubic NLW in a weighted Besov space $H^{-\varepsilon}(\rho)$ and establishing global invariance of the infinite-volume $\phi^4_2$ measure under NLW, along with a weaker weak invariance for NLS. The methodology relies on constructing the infinite-volume measure as a weak limit of periodic measures, proving tightness, and using Liouville-type arguments for truncated systems, then passing to the limit. The results extend the probabilistic solution theory for dispersive PDEs in higher dimensions and clarify the behavior of singular Gibbs measures under nonlinear evolution, with implications for stochastic quantization and $P(\phi)_2$-theory on unbounded domains.

Abstract

We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over $\mathbb R^2$. To achieve this, we show that any weak limit of $φ^4$ measures on increasing tori is invariant under the equation. We review and slightly simplify the periodic theory and the construction of the weak limit measure, and then use finite speed of propagation to reduce the infinite-volume case to the previous setup. Our argument also gives a weaker invariance result on the nonlinear Schrödinger equation in the same setting.

Invariance of $φ^4$ measure under nonlinear wave and Schrödinger equations on the plane

TL;DR

The work investigates invariance properties of the two-dimensional measure under nonlinear wave and Schrödinger dynamics on the plane. It combines stochastic quantization, Wick renormalization, and Besov-space techniques to transfer periodic invariance results to infinite volume via finite speed of propagation, yielding almost sure global well-posedness for the cubic NLW in a weighted Besov space and establishing global invariance of the infinite-volume measure under NLW, along with a weaker weak invariance for NLS. The methodology relies on constructing the infinite-volume measure as a weak limit of periodic measures, proving tightness, and using Liouville-type arguments for truncated systems, then passing to the limit. The results extend the probabilistic solution theory for dispersive PDEs in higher dimensions and clarify the behavior of singular Gibbs measures under nonlinear evolution, with implications for stochastic quantization and -theory on unbounded domains.

Abstract

We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over . To achieve this, we show that any weak limit of measures on increasing tori is invariant under the equation. We review and slightly simplify the periodic theory and the construction of the weak limit measure, and then use finite speed of propagation to reduce the infinite-volume case to the previous setup. Our argument also gives a weaker invariance result on the nonlinear Schrödinger equation in the same setting.
Paper Structure (23 sections, 73 theorems, 313 equations)

This paper contains 23 sections, 73 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. The phi^4 measure
  3. Previous literature and extensions
  4. Outline and notation
  5. Acknowledgements
  6. Besov spaces
  7. Stochastic quantization
  8. Gaussian free field
  9. Coupling of the phi^4 measure and the GFF
  10. Wick powers of phi42
  11. Invariance of periodic NLW
  12. Linear part
  13. Fixed-point iteration
  14. Globalization in time
  15. Invariance of non-truncated measure
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\vec{\mu}$ be the product of infinite-volume $\phi^4$ and white noise measures and fix $\varepsilon > 0$. Let $H^{-\varepsilon}(\rho)$ be the Besov space with a sufficiently integrable polynomial weight $\rho$. For $\vec{\mu}$-almost all initial data, the nonlinear wave equation eq:nlw has a un

Theorems & Definitions (163)

  • Theorem 1.1: Global existence and uniqueness
  • Theorem 1.2: Weak invariance of NLS
  • Remark 1.3
  • Example 1.4
  • Remark 2.1
  • Definition 2.2: Littlewood--Paley blocks
  • Definition 2.3: Weighted Besov space
  • Theorem 2.4: Product inequality
  • proof
  • Theorem 2.5: Duality
  • ...and 153 more