Gibbs States and Gibbsian Specifications on the space $\mathbb{R}^{\mathbb{N}}$

Abstract: We are interested in the study of Gibbs and equilbrium probabilities on the lattice $\mathbb{R}^{\mathbb{N}}$. Consider the unilateral full-shift defined on the non-compact set $\mathbb{R}^{\mathbb{N}}$ and an $\alpha$-H\"older continuous potential $A$ from $\mathbb{R}^{\mathbb{N}}$ into $\mathbb{R}$. From a suitable class of a priori probability measures $\nu$ (over the Borelian sets of $\mathbb{R}$) we define the Ruelle operator associated to $A$ (using an adequate extension of this operator to the compact set $\overline{\mathbb{R}}^\mathbb{N}=(S^1)^\mathbb{N}$) and we show the existence of eigenfunctions, conformal probability measures and equilibrium states associated to $A$. We are also able to show several of the well known classical properties of Thermodynamic Formalism for both of these probability measures. The above, can be seen as a generalization of the results obtained in the compact case for the XY-model. We also introduce an extension of the definition of entropy and show the existence of $A$-maximizing measures (via ground states for $A$); we show the existence of the zero temperature limit under some mild assumptions. Moreover, we prove the existence of an involution kernel for $A$ (this requires to consider the bilateral full-shift on $\mathbb{R}^{\mathbb{Z}}$). Finally, we build a Gibbsian specification for the Borelian sets on the set $\mathbb{R}^{\mathbb{N}}$ and we show that this family of probability measures satisfies a \emph{FKG}-inequality.