The Gibbs measures and partial differential equations. I. Ideas and local aspects
type: Item
instance of: scholarly article
title: The Gibbs measures and partial differential equations. I. Ideas and local aspects
zbMATH Open document ID: 0608.60093
DOI: 10.1007/BF01220997
author: Boguslaw Zegarliński
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: We investigate the connections of the Gibbs measures, which appear in Euclidean field theory, and the corresponding partial differential equations of classical Euclidean field theory. We show that for the quasilinear Dirichlet problems in the bounded regions \(\Lambda \subset {\mathbb{R}}^ d\) one can naturally associate a local specification \({\mathcal E}^ U=\{\pi^{\eta}_{\Lambda^ c}\}\) of a Gibbs measure theory. This ground local specification is defined as a family of probability measures \(\pi^{\eta}_{\Lambda^ c}\) concentrated on the solutions of the Dirichlet problems in \(\Lambda\) with boundary data \(\eta\) (respectively). Next we define a family \({\mathcal E}^ U_{\beta}=\{E^{\eta}_{\Lambda^{c,\beta}}\}\) of local specifications corresponding to a two-dimensional Euclidean field with interaction U and at temperature \(\beta^{-1}>0\). It is shown that \({\mathcal E}^ U_{\beta}\to {\mathcal E}^ U\) as \(\beta\to \infty\) in the sense that for any bounded region \(\Lambda \subset {\mathbb{R}}^ d\) (satisfying some technical conditions) \(E^{\eta}_{\Lambda^{c,\beta}}\to \pi^{\eta}_{\Lambda^ c}\) weakly as \(\beta\to \infty\), for \(\mu\)-a.e. tempered distributions \(\eta\in {\mathcal S}'({\mathbb{R}}^ 2)\) and any regular probability measure \(\mu\) on (\({\mathcal S}',{\mathcal B})\). (The class of the regular probability measures by its definition contains all the probability measures constructed in Euclidean field theory. It is known that the set of functions has measure zero for any regular \(\mu\).) In the last section the solutions of quasilinear elliptic Dirichlet problems with distributional boundary data \(\eta\) are shown to exist, \(\mu\)-a.e. for any regular \(\mu\) ; see also \textit{M. Röckner} and the author, ''The Dirichlet problem for quasi-linear partial differential operators with boundary data given by distribution.'' to appear in Proc. 3rd BiBoS Symp., Stochastic Processes Math. Phys., Bielefeld 1985, Lect. Notes Math.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 3984281
zbMATH Keywords: Gibbs measures
zbMATH Keywords: Euclidean field theory
zbMATH Keywords: Dirichlet problems
zbMATH Keywords: Euclidean field with interaction
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3984281