Invariant Gibbs measures for the 2-$d$ defocusing nonlinear wave equations

Tadahiro Oh; Laurent Thomann

doi:10.5802/afst.1620

Invariant Gibbs measures for the 2-

d

defocusing nonlinear wave equations

Tadahiro Oh ¹ ; Laurent Thomann ²

¹ School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
² Institut Élie Cartan, Université de Lorraine, B.P. 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 1-26.

Résumé
Abstract

On considère les équations des ondes non-linéaires défocalisantes sur le tore de dimension deux. On construit des mesures de Gribbs invariantes pour les équation renormalisés au sens de Wick. On prouve ensuite une propriété d’universalité faible pour ces équations renormalisés, en montrant qu’elle apparaissent comme limites d’équations d’ondes non renormalisées avec conditions initiales aléatoires de loi gaussienne.

We consider the defocusing nonlinear wave equations (NLW) on the two-dimensional torus. In particular, we construct invariant Gibbs measures for the renormalized so-called Wick ordered NLW. We then prove weak universality of the Wick ordered NLW, showing that the Wick ordered NLW naturally appears as a suitable scaling limit of non-renormalized NLW with Gaussian random initial data.

Reçu le : 2017-03-31
Accepté le : 2019-09-23
Publié le : 2020-07-24

DOI : 10.5802/afst.1620

Classification : 35L71
Mots clés : nonlinear wave equation, nonlinear Klein–Gordon equation, Gibbs measure, Wick ordering, Hermite polynomial, white noise functional, weak universality

Affiliations des auteurs :

Tadahiro Oh ¹ ; Laurent Thomann ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Tadahiro Oh; Laurent Thomann. Invariant Gibbs measures for the 2-$d$ defocusing nonlinear wave equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 1-26. doi : 10.5802/afst.1620. https://afst.centre-mersenne.org/articles/10.5802/afst.1620/

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