Statistical inference for the stochastic wave equation based on discrete observations

TL;DR

The paper addresses statistical inference for the wave speed $\vartheta$ in a high-dimensional stochastic wave equation driven by spatially colored Riesz noise, using discrete observations in space and time. It introduces a method-of-moments framework based on second-order space, time, and space-time variations, and proves central limit theorems with explicit Fejér-kernel representations that decouple spatial and temporal effects. The contributions include precise asymptotics for expectations and variances of the variations, several CLTs, and robust estimators $\hat{\vartheta}$ for the different observation schemes, with regime-dependent behavior governed by the sampling ratio $\alpha=\delta/\lambda$. These results provide practical, efficient inference tools for the wave speed in SPDEs and illuminate the interplay between discretization and hyperbolic dynamics in high dimensions.

Abstract

The wave speed of a stochastic wave equation driven by Riesz noise on the unbounded multidimensional spatial domain is estimated based on discrete measurements. Central limit theorems for second-order variations of the observations in space, time, and space-time are established. Under general assumptions on the spatial and temporal sampling frequencies, the resulting method-of-moments estimators are asymptotically normally distributed. The covariance structure of the discrete increments admits a closed-form representation involving two different Fejér-type kernels, enabling a precise analysis of the interplay between spatial and temporal contributions.

Figures (5)

• Figure 1: Illustration of the stochastic wave equation in $d=1$ with $\vartheta=\beta=0.5$, and non-zero initial condition together with the space-time observation grid with $n=10$ spatial and $m=10$ temporal observation points. The distance between two adjacent time points and spatial points is denoted by $\delta$ and $\lambda$, respectively. The simulation is based on a semi-implicit Euler scheme on a large bounded domain.
• Figure 3: Visualisation of the spatial observation points $(x_k)_{k=0}^{n+1}$ for a fixed $n \in \mathbb{N}$, $\lambda \in (0,1)$ in $d=2$.
• Figure 4: Visualisation of the Fejér kernel $\mathfrak{F}_{\mathrm{sp}}$ defined through \ref{['eq:introducing_the_fejér_kernel']} for different values of $n \in \mathbb{N}$.
• Figure 5: Visualisation of the time discrete observations $t_0, t_1, \dots, t_{m+1}$ with distance $t_{i+1}-t_{i}=\delta$ in $d=2$ at the fixed spatial point $x=(x_1, x_2)$.
• Figure 6: Visualisation of the space-time grid $(t_i, x_k)$ in $d=2$ for $i=0, \dots, m+1$ and $k=0, \dots, n+1$. The blue vector symbolising the direction of the spatial measurements is $\rho \in \mathbb{S}^1$.

Theorems & Definitions (81)

• Proposition 2.1
• Remark 2.2
• Remark 3.1
• Lemma 3.2
• Remark 3.3
• Proposition 3.4
• Remark 3.5
• Proposition 3.6
• Remark 3.7
• Remark 3.8