Parameter estimation in hyperbolic linear SPDEs from multiple measurements

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.