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Well-posedness of a gauge-covariant wave equation with space-time white noise forcing

Bjoern Bringmann, Igor Rodnianski

TL;DR

The paper addresses probabilistic global well-posedness for a gauge-covariant wave equation in 1+2 dimensions forced by space-time white noise, working in the Lorenz gauge. It develops a novel Ansatz that decomposes the scalar field into a random linear component and a smoother remainder, and constructs random operators to capture high-high, high-low, and low-high interactions, using resolvent estimates and lattice-point counting to handle non-smoothing nonlinearities. A renormalization with a time-dependent mass is implemented to control the divergent quadratic term in the vector potential, and the truncated solutions are shown to converge almost surely to a global solution, yielding probabilistic global well-posedness. In addition, the authors prove a probabilistic null-form failure which exposes a potential obstruction to extending these methods to the stochastic Maxwell-Klein-Gordon equation, highlighting a fundamental limitation and guiding future work in stochastic geometric wave equations.

Abstract

We first introduce a new model for a two-dimensional gauge-covariant wave equation with space-time white noise. In our main theorem, we obtain the probabilistic global well-posedness of this model in the Lorenz gauge. Furthermore, we prove the failure of a probabilistic null-form estimate, which exposes a potential obstruction towards the probabilistic well-posedness of a stochastic Maxwell-Klein-Gordon equation.

Well-posedness of a gauge-covariant wave equation with space-time white noise forcing

TL;DR

The paper addresses probabilistic global well-posedness for a gauge-covariant wave equation in 1+2 dimensions forced by space-time white noise, working in the Lorenz gauge. It develops a novel Ansatz that decomposes the scalar field into a random linear component and a smoother remainder, and constructs random operators to capture high-high, high-low, and low-high interactions, using resolvent estimates and lattice-point counting to handle non-smoothing nonlinearities. A renormalization with a time-dependent mass is implemented to control the divergent quadratic term in the vector potential, and the truncated solutions are shown to converge almost surely to a global solution, yielding probabilistic global well-posedness. In addition, the authors prove a probabilistic null-form failure which exposes a potential obstruction to extending these methods to the stochastic Maxwell-Klein-Gordon equation, highlighting a fundamental limitation and guiding future work in stochastic geometric wave equations.

Abstract

We first introduce a new model for a two-dimensional gauge-covariant wave equation with space-time white noise. In our main theorem, we obtain the probabilistic global well-posedness of this model in the Lorenz gauge. Furthermore, we prove the failure of a probabilistic null-form estimate, which exposes a potential obstruction towards the probabilistic well-posedness of a stochastic Maxwell-Klein-Gordon equation.
Paper Structure (22 sections, 27 theorems, 264 equations)

This paper contains 22 sections, 27 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. A gauge-covariant wave equation with space-time white noise
  3. Gauge-invariance
  4. Continuity equation and random currents
  5. Lorenz gauge
  6. Renormalization
  7. Main results
  8. Maxwell-Klein-Gordon and the failure of a probabilistic null-form estimate
  9. Ansatz
  10. Notation and preliminaries
  11. General notation
  12. Probability theory
  13. Harmonic analysis
  14. $X^{s,b}$-spaces and tensors
  15. Lattice point counting estimates
  16. ...and 7 more sections

Key Result

Theorem 1.3

Let $d=2$, let $\delta>0$, and let $(\phi_0,\phi_1) \in \mathscr{H}_x^{1/4}(\mathbb{T}^2)$. Furthermore, let $\zeta$ be a complex-valued space-time white noise, let $(J^\alpha)_{\alpha=0}^2$ be a space-time white noise current, and assume that $\zeta$ and $(J^\alpha)_{\alpha=0}^2$ are probabilistic almost surely exists in for all $T>0$.

Theorems & Definitions (71)

  • Definition 1.1: Space-time white noise current
  • Remark 1.2: Random vector potential
  • Theorem 1.3: Probabilistic global well-posedness
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Failure of a probabilistic null-form estimate
  • Remark 1.10
  • ...and 61 more