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This is a Quantitative Study study.

October 5, 2025Open Access

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

Key Points

The local linearization method effectively estimates the diffusion parameter in nonlinear wave equations, and reveals significant insights into solution behavior.
Key findings demonstrate how the second-order increments align with linearized equations, suggesting strong connections between noise dynamics and solution properties.
The approach utilizes a Gaussian noise model with a Riesz-kernel covariance structure, indicating potential robustness across various models and applications.
Applications include analyzing quadratic variation, providing a pathway to estimate the diffusion parameter consistently.

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. 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.

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