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


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 \(\mathcal {E}^{\mathbb {K}}\).


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

Mathematical Subject Classification (2010)

60J60 60G57 


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  1. 1.
    Alberverio, S., Kondratiev, Yu.G., Röckner, M.: Analysis and geometry on configuration spaces. The Gibbsian case. J Func.Anal. 157, 242–291 (1998)CrossRefGoogle Scholar
  2. 2.
    Boothby, W.M.: An Introduction to differentiable manifolds and Riemannian geometry. Academic Press, San Diego (1975)MATHGoogle Scholar
  3. 3.
    Daley, D. J., Vere-Jones, D.: An introduction to the theory of point processes. Vol. II. General theory and structure. Second edition, Springer, New York (2008)Google Scholar
  4. 4.
    Dynkin, E.B.: Markov Processes. Springer, Berlin (1965)CrossRefMATHGoogle Scholar
  5. 5.
    Fukushima, M.: Dirichlet Forms and Symmetric Markov Processes. North-Holland, Amsterdam (1980)Google Scholar
  6. 6.
    Hagedorn, D., Kondratiev, Y., Pasurek, T., Röckner, M.: Gibbs states over the cone of discrete measures. J. Funct. Anal. 264, 2550–2583 (2013)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Hagedorn, D., Kondratiev, Y., Lytvynov, E., Vershik, A.: Laplace operators in gamma analysis, arXiv:1411.0162, to appear in Trends of Mathematics, Birkhäuser.
  8. 8.
    Kallenberg, O.: Random measures. Fourth edition. Akademie-Verlag, Berlin Academic Press, London (1986)Google Scholar
  9. 9.
    Kingman, J.F.C.: Completely random measures. Pacific. J. Math 21, 59–78 (1967)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Kondratiev, Y., Lytvynov, E., Vershik, A.: Laplace operators on the cone of Radon measures, to appear in J. Funct. Anal.Google Scholar
  11. 11.
    Kondratiev, Y., Lytvynov, Röckner, M.: Infinite interacting diffusion particles I: Equilibrium process and its scaling limit. Forum Math. 18, 9–43 (2006)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Kuratowski, K.: Topology. Vol. I. Academic Press, New York–London Warsaw (1966)Google Scholar
  13. 13.
    Ma, Z.-M., Röckner, M.: An Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin (1992)CrossRefGoogle Scholar
  14. 14.
    Ma, Z.-M., Röckner, M.: Construction of diffusions on configuration spaces. Osaka. J. Math 37, 273–314 (2000)MATHGoogle Scholar
  15. 15.
    Nguyen, X.X., Zessin, H.: Integral and differentiable characterizations of the Gibbs process. Math. Nachr. 88, 105–115 (1979)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Putan, D.: Uniqueness of equilibrium states of some models of interacting particle systems. PhD Thesis, Universität Bielefeld, Bielefeld, available at (2014)
  17. 17.
    Röckner, M., Schmuland, B.: Quasi-regular Dirichlet forms: examples and counterexamples. Canad. J. Math. 47, 165–200 (1995)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Röckner, M., Schmuland, B.: A support property for infinite-dimensional interacting diffusion processes. C. R. Acad. Sci. Paris Sér. I Math 326, 359–364 (1998)CrossRefMATHGoogle Scholar
  19. 19.
    Tsilevich, N., Vershik, A., Yor, M.: An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process. J. Funct. Anal 185, 274–296 (2001)CrossRefMathSciNetMATHGoogle Scholar

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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