Given any measurable subset of a closed Riemannian manifold and given any , we define as the smallest average time over spent by all geodesic rays in . Our first main result, which is of geometric nature, states that, under regularity assumptions, is the maximal possible discrepancy of when taking the closure. Our second main result is of probabilistic nature: considering a regular checkerboard on the flat two-dimensional torus made of square white cells, constructing random subsets by darkening cells randomly with a probability , we prove that the random law converges in probability to as . We discuss the consequences in terms of observability of the wave equation.
Étant donné un sous-ensemble mesurable d’une variété riemannienne compacte et étant donné , on définit comme étant le plus petit temps moyen passé par les rayons géodésiques dans . Notre premier résultat principal, qui est de nature géométrique, établit que, sous des conditions de régularité, peut augmenter au maximum de en passant à l’adhérence. Notre second résultat principal est de nature probabiliste : considérant un damier régulier sur le tore plat de dimension formé de carrés blancs, construisant des ensembles aléatoires en noircissant les carrés de manière aléatoire avec probabilité , on montre que la variable aléatoire converge en probabilité vers lorsque . Nous discutons les conséquences en termes d’observabilité de l’équation des ondes.
Accepted:
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Keywords: Observability, wave equation, Riemannian geometry, random set
Mot clés : Observabilité, équation des ondes, géométrie riemannienne, ensemble aléatoire
Emmanuel Humbert 1; Yannick Privat 2; Emmanuel Trélat 3
@article{JEP_2022__9__431_0, author = {Emmanuel Humbert and Yannick Privat and Emmanuel Tr\'elat}, title = {Geometric and probabilistic results for the~observability of the wave equation}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {431--461}, publisher = {\'Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.186}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.186/} }
TY - JOUR AU - Emmanuel Humbert AU - Yannick Privat AU - Emmanuel Trélat TI - Geometric and probabilistic results for the observability of the wave equation JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 431 EP - 461 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.186/ DO - 10.5802/jep.186 LA - en ID - JEP_2022__9__431_0 ER -
%0 Journal Article %A Emmanuel Humbert %A Yannick Privat %A Emmanuel Trélat %T Geometric and probabilistic results for the observability of the wave equation %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 431-461 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.186/ %R 10.5802/jep.186 %G en %F JEP_2022__9__431_0
Emmanuel Humbert; Yannick Privat; Emmanuel Trélat. Geometric and probabilistic results for the observability of the wave equation. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 431-461. doi : 10.5802/jep.186. https://jep.centre-mersenne.org/articles/10.5802/jep.186/
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