Geometric and probabilistic results for the observability of the wave equation

Given any measurable subset $\omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $\omega$ as an observation subset: the condition $\ell^T(\omega)>0$ is the well known \emph{Geometric Control Condition}. In this article we establish two properties of the functional $\ell^T$, one is geometric and the other is probabilistic. The first geometric property is on the maximal discrepancy of $\ell^T$ when taking the closure. We may have $\ell^T(\mathring{\omega})<\ell^T(\overline\omega)$ whenever there exist rays grazing $\omega$ and the discrepancy between both quantities may be equal to $1$ for some subsets $\omega$. We prove that, if the metric $g$ is $C^2$ and if $\omega$ satisfies a slight regularity assumption, then $\ell^T(\overline\omega) \leq \frac{1}{2} \left( \ell^T(\mathring{\omega}) + 1 \right)$. We also show that our assumptions are essentially sharp; in particular, surprisingly the result is wrong if the metric $g$ is not $C^2$. As a consequence, if $\omega$ is regular enough and if $\ell^T(\overline\omega)>1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $\omega$ in time $T$. The second property is of probabilistic nature. We take $M=\mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random subsets $\omega_\varepsilon^n$ by darkening each cell in this grid with a probability $\varepsilon$. We prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$. As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $\omega_\varepsilon^n$ in time $T$.