Local linearization for estimating the diffusion parameter of nonlinear stochastic wave equations with spatially correlated noise

TL;DR

The paper develops a bi-parameter local linearization framework for the one-dimensional nonlinear stochastic wave equation with spatially colored noise, establishing that second-order increments behave like those of a linearized equation modulated by the diffusion function $F(u)$ and quantifying the remainder with rate $\varepsilon^{2H+1/2}$. This enables a consistent, rotation-based quadratic-variation approach to estimate the diffusion parameter $\theta$ from spatiotemporal data, via a limit $Q_N(v_{\theta}) \to \theta^2 2^{H-5/2}\iint F^2(v_{\theta}) \,d\tau d\lambda$ and a practical estimator $\hat{\theta}_N$. Theoretical results are complemented by numerical experiments under space-time white and time-white/space-colored noise, confirming consistency and revealing finite-sample behavior. The work extends prior white-noise results to spatially colored noise, broadening applicability to physically realistic noise structures in stochastic wave phenomena.

Abstract

We study the bi-parameter local linearization of the one-dimensional nonlinear stochastic wave equation driven by a Gaussian noise, which is white in time and has a spatially homogeneous covariance structure of Riesz-kernel type. We establish that the second-order increments of the solution can be approximated by those of the corresponding linearized wave equation, modulated by the diffusion coefficient. These findings extend the previous results of Huang et al. \cite{HOO2024}, which addressed the case of space-time white noise. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

Figures (6)

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Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Proposition 2.1
• Lemma 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm error 1']}
• Corollary 4.1
• proof