Local linearization for the nonlinear damped stochastic Klein-Gordon equation

TL;DR

This paper demonstrates that the second-order increments of solutions to the nonlinear damped stochastic Klein-Gordon equation can be approximated by those of its linearized version, aiding in parameter estimation and analysis.

Contribution

It extends previous linearization results to the nonlinear damped stochastic Klein-Gordon equation, overcoming complex Green function structures with new analytical estimates.

Findings

Second-order increments approximated by linearized equation

Constructed a consistent estimator for the diffusion parameter

Analyzed quadratic variation of the solution

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.