Inspired by the work of Zhidkov on the KdV equation, we perform a construction of weighted Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. The resulting measures are supported by Sobolev spaces of increasing regularity. We also prove a property on the support of these measures leading to the conjecture that they are indeed invariant by the flow of the Benjamin-Ono equation.
Inspirés par le travail de Zhidkov sur l'équation KdV, nous construisons des mesures gaussiennes à poids associées à une loi de conservation arbitraire de l'équation de Benjamin-Ono. Les supports de ces mesures sont constitués de fonctions de régularité de Sobolev croissantes. On démontre aussi une propriété-clé des mesures qui nous conduit à conjecturer leur invariance par le flot de l'équation.
Keywords: dispersive equations, Wiener chaos, invariant measures
Mot clés : Équations dispersives, chaos de Wiener, mesures invariantes
@article{ASENS_2013_4_46_2_249_0, author = {Tzvetkov, Nikolay and Visciglia, Nicola}, title = {Gaussian measures associated to the higher order conservation laws of the {Benjamin-Ono} equation}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {249--299}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 46}, number = {2}, year = {2013}, doi = {10.24033/asens.2189}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2189/} }
TY - JOUR AU - Tzvetkov, Nikolay AU - Visciglia, Nicola TI - Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation JO - Annales scientifiques de l'École Normale Supérieure PY - 2013 SP - 249 EP - 299 VL - 46 IS - 2 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2189/ DO - 10.24033/asens.2189 LA - en ID - ASENS_2013_4_46_2_249_0 ER -
%0 Journal Article %A Tzvetkov, Nikolay %A Visciglia, Nicola %T Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation %J Annales scientifiques de l'École Normale Supérieure %D 2013 %P 249-299 %V 46 %N 2 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2189/ %R 10.24033/asens.2189 %G en %F ASENS_2013_4_46_2_249_0
Tzvetkov, Nikolay; Visciglia, Nicola. Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 2, pp. 249-299. doi : 10.24033/asens.2189. http://www.numdam.org/articles/10.24033/asens.2189/
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