GEOMETRIC AND PROBABILISTIC RESULTS FOR THE OBSERVABILITY OF THE WAVE EQUATION

Given any measurable subset omega of a closed Riemannian manifold and given any T > 0, we define l(T) (omega) is an element of [0, 1] as the smallest average time over [0, T] spent by all geodesic rays in omega. Our first main result, which is of geometric nature, states that, under regularity assumptions, 1/2 is the maximal possible discrepancy of l(T) when taking the closure. Our second main result is of probabilistic nature: considering a regular checkerboard on the flat two-dimensional torus made of n(2) square white cells, constructing random subsets omega(n)(epsilon) by darkening cells randomly with a probability epsilon, we prove that the random law l(T )(omega(n)(epsilon)) converges in probability to epsilon as n -> +infinity. We discuss the consequences in terms of observability of the wave equation.