Gibbs measure over the cone of vector-valued discrete measures

We consider a gas whose each particle is characterised by a pair $(x,v_x)$ with the position $x\in {\mathbb{R}}^d$ and the velocity $v_x\in {\mathbb{R}}_0^d={\mathbb{R}}^d\setminus \{0\}$. We define Gibbs measures on the cone of vector-valued measures and aim to prove their existence. We introduce the family of probability measures ${\mu }_{\lambda }$ on the cone $\mathbb{K}({\mathbb{R}}^d)$. We define local Hamiltonian and partition functions for a positive, symmetric, bounded and measurable pair potential. Using those above, we define Gibbs’s measure as a solution to the Dobrushin-Lanford-Ruelle equation. In particular, we focus on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is non-empty and relatively compact.