Non-parametric estimation for the stochastic wave equation

TL;DR

This paper develops a non-parametric estimation method for the wave speed in a stochastic wave equation driven by space-time white noise, establishing asymptotic properties of the estimator based on local observations.

Contribution

It introduces a novel local observation scheme and proves asymptotic normality of an augmented maximum likelihood estimator for the wave speed.

Findings

Asymptotic normality of the estimator as observation resolution increases

Intrinsic relation between Fisher information and kinetic energy

Proof of an asymptotic energy equipartition principle

Abstract

The spatially dependent wave speed of a stochastic wave equation driven by space-time white noise is estimated using the local observation scheme. Given a fixed time horizon, we prove asymptotic normality for an augmented maximum likelihood estimator as the resolution level of the observations tends to zero. We show that the expectation and variance of the observed Fisher information are intrinsically related to the kinetic energy within an associated deterministic wave equation and prove an asymptotic equipartition of energy principle using the notion of asymptotic Riemann-Lebesgue operators.