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Potential Analysis

, Volume 44, Issue 1, pp 71–90 | Cite as

Equilibrium Diffusion on the Cone of Discrete Radon Measures

  • Diana Conache
  • Yuri G. Kondratiev
  • Eugene Lytvynov
Article
  • 78 Downloads

Abstract

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form
$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$
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 \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

Keywords

Completely random measure Diffusion process Discrete radon measure Dirichlet form Gibbs measure 

Mathematical Subject Classification (2010)

60J60 60G57 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Diana Conache
    • 1
  • Yuri G. Kondratiev
    • 1
    • 2
  • Eugene Lytvynov
    • 3
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.National Pedagogical Dragomanov UniversityKyivUkraine
  3. 3.Department of Mathematics, Swansea UniversitySingleton ParkSwanseaUK

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