Gibbs measure over the cone of vector-valued discrete measures - Di Persio, Luca; Kondratiev, Yuri; Vardanyan , Viktorya (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.9 We consider a gas each particle of which is characterized by a pair (x,v_x), where xR^d is the position and v_xR^d_0=R^d{0} is the velocity. Gibbs measures are defined on the cone of vector-valued measures. Our aim is to prove their existence. We introduce a family of probability measures μ_λ on the cone K(R^d) and define local Hamiltonian and partition functions for a positive, symmetric, bounded, and measurable pair potential. By using the definitions mentioned above, we define Gibbs 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 nonempty and relatively compact.