Tohoku Mathematical Journal
2019
March
SECOND SERIES VOL. 71, NO. 1

Tohoku Math. J.
71 (2019), 9-33

Title INFINITE PARTICLE SYSTEMS OF LONG RANGE JUMPS WITH LONG RANGE INTERACTIONS

Author Syota Esaki

(Received December 7, 2015, revised July 26, 2016)
Abstract. 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.

Mathematics Subject Classification. Primary 60K35; Secondary 60J75.
Key words and phrases. Interacting Lévy processes, infinitely particle systems, Dirichlet form, jump type, logarithmic potential.

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