The Gibbs measures and partial differential equations. I. Ideas and local aspects
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.
scientific article; zbMATH DE number 3984281
scientific article