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Propagation of Singularities for the Damped Stochastic Klein-Gordon Equation

Hongyi Chen, Cheuk Yin Lee

TL;DR

We address regularity and the propagation of stochastic singularities for the 1+1 dimensional damped stochastic Klein-Gordon equation driven by space-time white noise. The authors reduce to the critically damped model by decomposing the solution into a critical part $u_C$ and a Lipschitz remainder $u_L$, proving the main LIL and modulus-of-continuity results for $u_C$ and transferring them to the full equation. They establish law-of-the-iterated-logarithm and Levy's modulus of continuity along characteristic directions, and show that random LIL singularities exist and propagate along the complementary characteristic, mirroring known phenomena for stochastic wave equations. The work builds a bridge to microlocal analysis by discussing wavefront-set-type descriptions and conjecturing β-MC/LIL wavefront sets, thereby strengthening the connection between stochastic regularity and propagation of singularities in SPDEs.

Abstract

For the $1+1$ dimensional damped stochastic Klein-Gordon equation, we show that random singularities associated with the law of the iterated logarithm exist and propogate in the same way as the stochastic wave equation. This provides evidence for possible connections to microlocal analysis, ie. the exact regularity and singularities described in this paper should admit wavefront set type descriptions whose propagation is determined by the highest order terms of the linear operator. Despite the results being exactly the same as those of the wave equation, our proofs are significantly different than the proofs for the wave equation. Miraculously, proving our results for the critically damped equation implies them for the general equation, which significantly simplifies the problem. Even after this simplification, many important parts of the proof are significantly different than (and we think are more intuitive from the PDE viewpoint compared to) existing proofs for the wave equation.

Propagation of Singularities for the Damped Stochastic Klein-Gordon Equation

TL;DR

We address regularity and the propagation of stochastic singularities for the 1+1 dimensional damped stochastic Klein-Gordon equation driven by space-time white noise. The authors reduce to the critically damped model by decomposing the solution into a critical part and a Lipschitz remainder , proving the main LIL and modulus-of-continuity results for and transferring them to the full equation. They establish law-of-the-iterated-logarithm and Levy's modulus of continuity along characteristic directions, and show that random LIL singularities exist and propagate along the complementary characteristic, mirroring known phenomena for stochastic wave equations. The work builds a bridge to microlocal analysis by discussing wavefront-set-type descriptions and conjecturing β-MC/LIL wavefront sets, thereby strengthening the connection between stochastic regularity and propagation of singularities in SPDEs.

Abstract

For the dimensional damped stochastic Klein-Gordon equation, we show that random singularities associated with the law of the iterated logarithm exist and propogate in the same way as the stochastic wave equation. This provides evidence for possible connections to microlocal analysis, ie. the exact regularity and singularities described in this paper should admit wavefront set type descriptions whose propagation is determined by the highest order terms of the linear operator. Despite the results being exactly the same as those of the wave equation, our proofs are significantly different than the proofs for the wave equation. Miraculously, proving our results for the critically damped equation implies them for the general equation, which significantly simplifies the problem. Even after this simplification, many important parts of the proof are significantly different than (and we think are more intuitive from the PDE viewpoint compared to) existing proofs for the wave equation.

Paper Structure

This paper contains 12 sections, 16 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Literature review and motivation
  3. Paper organization and notation
  4. Preliminaries
  5. Reduction to the critical damping case
  6. Reduction of Theorem \ref{['th:lil']}
  7. Reduction of Theorem \ref{['th:mc']}
  8. Reduction of Theorem \ref{['th:sim:lil']}
  9. Reduction of Theorem \ref{['th:sing']}
  10. Proof of Theorems \ref{['th:lil']} and \ref{['th:mc']}
  11. Proof of Theorem \ref{['th:sim:lil']}
  12. Proof of Theorem \ref{['th:sing']}

Key Result

Theorem 1.1

For any fixed $t_0\ge0$ and $x_0 \in \mathbb{R}$, there exists a constant $0<K_1<\infty$ depending on $(t_0, x_0)$ such that

Theorems & Definitions (34)

  • Theorem 1.1: Law of the iterated logarithm
  • Theorem 1.2: Modulus of continuity
  • Theorem 1.3
  • Theorem 1.4: Existence and propagation of singularities
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 24 more