Parameter estimation in hyperbolic linear SPDEs from multiple measurements

TL;DR

This paper develops a statistical framework for estimating coefficients in hyperbolic SPDEs from multiple measurements, establishing asymptotic normality and convergence rates for the estimators, with insights into the Fisher information matrix.

Contribution

It introduces a novel asymptotic analysis of maximum likelihood estimators for hyperbolic SPDEs using local measurements, including the behaviour of operator functions and Fisher information.

Findings

Asymptotic normality of estimators established as measurement resolution improves.

Convergence rates for dissipative coefficients match those in parabolic problems.

Additional smoothing properties of operator functions are characterized using functional calculus.

Abstract

The coefficients of elastic and dissipative operators in a linear hyperbolic SPDE are jointly estimated using multiple spatially localised measurements. As the resolution level of the observations tends to zero, we establish the asymptotic normality of an augmented maximum likelihood estimator. The rate of convergence for the dissipative coefficients matches rates in related parabolic problems, whereas the rate for the elastic parameters also depends on the magnitude of the damping. The analysis of the observed Fisher information matrix relies upon the asymptotic behaviour of rescaled $M, N$-functions generalising the operator cosine and sine families appearing in the undamped wave equation. In contrast to the energetically stable undamped wave equation, the $M, N$-functions emerging within the covariance structure of the local measurements have additional smoothing properties similar to the heat kernel, and their asymptotic behaviour is analysed using functional calculus.