# Regularity of the diffusion coefficient matrix for the lattice gas with energy

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doi:10.1016/j.anihpb.2004.03.006 Ann. I. H. Poincaré – PR 41 (2005) 45–67 D Abstrac In thi obtain t 2004 Résumé Dans obtient 2004 1. Intr In o equatio In th diffusio ticity o unique The Olla, a Bernou It se not hav E-ma 0246-020 doi:10.10 www.elsevier.com/locate/anihpb Regularity of the diffusion coefficient matrix for the lattice gas with energy Yukio Nagahata epartment of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531, Japan Received 5 April 2003; received in revised form 4 November 2003; accepted 30 March 2004 Available online 11 September 2004 t s paper we obtain the smoothness of the diffusion coefficient matrix for the lattice gas with energy. Furthermore we also he smoothness of the central limit theorem variances for certain functions. Elsevier SAS. All rights reserved. cet article, on montre la régularité de la matrice des coefficients de diffusion pour le gaz sur réseau avec énergie. On également la régularité des variances associées à certaines fonctions par le théorème limite central. Elsevier SAS. All rights reserved. oduction ur previous paper [6] we have introduced a lattice gas with energy and derived the fluctuation dissipation n for it. In this paper we prove that the diffusion coefficient matrix appearing in the equation is smooth. e derivation of hydrodynamic limit, uniqueness of the Cauchy problem of the weak solution of limiting n equation is needed. It seems unsolved in the existing literatures. But once smoothness and uniform ellip- f the diffusion coefficient matrix is established and if there exists a Lipschitz continuous solution, then the ness question is resolved. smoothness of the self-diffusion coefficient of the symmetric simple exclusion process is proved by Landim, nd Varadhan [4], and the smoothness of the diffusion coefficient for a lattice gas reversible under the lli measures is proved by Bernardin [1]. ems difficult to adapt to our model the method which intr

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