Gibbs states and Gibbsian specifications on the space ℝ^ℕ.

We are interested in the study of Gibbs and equilibrium probabilities on the space state R N . We consider the unilateral full-shift σ defined on the non-compact set R N , that is σ (x 1 , x 2 ,.. , x n ,..) = (x 2 , x 3 ,.. , x n ,..) , and a Hölder continuous potential A : R N → R . From a suitable class of a priori probability measures ν we define the Ruelle operator associated to A and we show the existence of eigenfunctions, conformal probability measures and equilibrium states associated to A. Moreover, we prove the existence of an involution kernel for A, we build a Gibbsian specification for the Borelian sets on R N and we show that this family of probabilities satisfies a FKG-inequality. [ABSTRACT FROM AUTHOR]