Tohoku Mathematical Journal

Infinite particle systems of long range jumps with long range interactions

Syota Esaki

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In this paper a general theorem for constructing infinite particle systems of jump type with long range interactions is presented. It can be applied to the system that each particle undergoes an $\alpha$-stable process and interaction between particles is given by the logarithmic potential appearing random matrix theory or potentials of Ruelle's class with polynomial decay. It is shown that the system can be constructed for any $\alpha \in (0, 2)$ if its equilibrium measure $\mu$ is translation invariant, and $\alpha$ is restricted by the growth order of the 1-correlation function of the measure $\mu$ in general case.

Article information

Tohoku Math. J. (2), Volume 71, Number 1 (2019), 9-33.

First available in Project Euclid: 9 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J75: Jump processes

interacting Lévy processes infinitely particle systems Dirichlet form jump type logarithmic potential


Esaki, Syota. Infinite particle systems of long range jumps with long range interactions. Tohoku Math. J. (2) 71 (2019), no. 1, 9--33. doi:10.2748/tmj/1552100440.

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  • S. Esaki and H.Tanemura, Infinite stochastic differential equation of interacting particle systems of long range jumps with long range interactions, in preparation.
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, Walter de Gruyter, 2010.
  • Y. Kondratiev, E. Lytvynov and M. Röckner, Equilibrium Kawasaki dynamics of continuous particle systems, Infin. Dimen. Anal. Quant. Prob. Rel. Top. 10 (2007), no.2, 185–209.
  • S. Kusuoka, Dirichlet forms and diffusion processes on Banach spaces, J. Fac. Sci. Univ. Tokyo. Sect. IA Math. 29 (1982), 79–95.
  • T.M. Liggett, The Stochastic Evolution of Infinite Systems of Interacting Particles. École d'Été de Probabilités de Saint-Flour, VI–1976, pp. 187–248. Lecture Notes in Math. Vol. 598, Springer-Verlag, Berlin, 1977.
  • T.M. Liggett, Interacting Particle Systems, Springer, New York, 1985.
  • E. Lytvynov and N. Ohlerich, A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes, Methods Funct. Anal. Topology 14 (2008), no. 1, 67–80.
  • Z.-M. Ma and M. Röckner, Introduction to the theory of (non-symmetric) Dirichlet forms. Springer-Verlag, Berlin, 1992.
  • H. Osada, Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions, Comm. Math. Phys. 176 (1996), 117–131.
  • H. Osada, Infinite-dimensional stochastic differential equations related to random matrices, Probab. Theory Related Fields 153 (2012), 471–509.
  • H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Ann. Probab. 41 (2013), 1–49.
  • H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field, Stochastic Process. Appl. 123 (2013), 813–838.
  • H. Osada and T. Shirai, Absolute continuity and singularity of Palm measures of the Ginibre point process, Probability Theory and Related Fields (Published on line)DOI 10.1007/s00440-015-0644-6.
  • H. Osada and H. Tanemura, Cores of Dirichlet forms related to Random Matrix Theory, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), 145–150.
  • D. Ruelle, Superstable interactions in classical statistical mechanics, Commun. Math. Phys. 18 (1970), 127–159.
  • B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), 377–385.
  • F. Spitzer, Random Processes Defined Through the Interaction of an Infinite Particle System, Springer Lecture Note in Mathematics, Vol. 89, 201–223, Springer Berlin Heidelberg, 1969.
  • H.Tanemura, Ergodicity for an infinite particle system in $\mathbb{R}^d$ of jump type with hard core interaction, J. Math. Soc. Japan, 41 (1989), no. 4, 681–697.
  • T. Uemura, On some path properties of symmetric stable-like processes for one dimension, Potential Anal. 16 (2002), 79–91.