Local linearization for the nonlinear damped stochastic Klein-Gordon equation

TL;DR

This work establishes a bi-parameter local linearization for the 1+1 dimensional nonlinear damped stochastic Klein-Gordon equation with multiplicative noise by analyzing the Green function in rotated characteristic coordinates. The authors prove that second-order increments of the nonlinear solution, after appropriate diffusion scaling, mimic those of the linearized equation, extending prior results for the stochastic wave equation to the Klein-Gordon setting with damping and mass. Central to the analysis are refined Green function estimates and careful domain decompositions that control drift and diffusion increment errors, enabling a tractable quadratic-variation analysis. As practical outcomes, the paper derives a convergence result for the quadratic variation and constructs a consistent estimator for the diffusion parameter, providing a statistical tool for SPDEs with multiplicative noise in hyperbolic-type models.

Abstract

For the $1+1$ dimensional nonlinear damped stochastic Klein-Gordon equation driven by space-time white noise, we prove that the second-order increments of the solution can be approximated, after scaling with the diffusion coefficient, by those of the corresponding linearized stochastic Klein-Gordon equation. This extends the result of Huang et al. \cite{HOO2024} for the stochastic wave equation. A key difficulty arises from the more complex structure of the Green function, which we overcome by means of subtle analytical estimates. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

Figures (3)

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Theorems & Definitions (19)

• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• proof : Proof of Theorem \ref{['thm error 1']}
• Lemma 3.1
• proof