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

Oh, Tadahiro; Thomann, Laurent

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

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.

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

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