2015/03/13 by Diana Conache, Conache, Diana, Yuri G. Kondratiev +3
Computer Science · Mathematics · #60G57 #60J60 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60G57 #msc:60J60
paper · pdf · doi:10.48550/arxiv.1503.04166
arxiv created 2015/03/13 · openalex publication_date 2015/03/13 · arxiv updated 2015/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbb K(\mathbb Rd) denote the cone of discrete Radon measures on \mathbb Rd. There is a natural differentiation on \mathbb K(\mathbb Rd): for a differentiable function F:\mathbb K(\mathbb Rd)→\mathbb R, one defines its gradient ∇\mathbb K F as a vector field which assigns to each η∈ \mathbb K(\mathbb Rd) an element of a tangent space Tη(\mathbb K(\mathbb Rd)) to \mathbb K(\mathbb Rd) at point η. Let ϕ:\mathbb Rd×\mathbb Rd→\mathbb R be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \mathbb Rd. In particular, μ is a probability measure on \mathbb K(\mathbb Rd) such that the set of atoms of a discrete measure η∈\mathbb K(\mathbb Rd) is μ-a.s. dense in \mathbb Rd. We consider the corresponding Dirichlet form \mathscr E\mathbb K(F,G)=∫\mathbb K(\mathbb Rd)⟨∇\mathbb K F(η), ∇\mathbb K G(η)⟩Tη(\mathbb K) dμ(η). Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d≥2, there exists a conservative diffusion process on \mathbb K(\mathbb Rd) which is properly associated with the Dirichlet form \mathscr E\mathbb K.