Tohoku Mathematical Journal
2019
September
SECOND SERIES VOL. 71, NO. 3

Tohoku Math. J.
71 (2019), 359-396

Title CANCELLATION OF FLUCTUATION IN STOCHASTIC RANKING PROCESS WITH SPACE-TIME DEPENDENT INTENSITIES

Author Tetsuya Hattori

(Received May 8, 2017, revised June 15, 2017)
Abstract. We consider the stochastic ranking process with space-time dependent unbounded jump rates for the particles. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution in the infinite particle limit. We assume topology of weak convergence for the space of distributions, which implies that the fluctuations among particles with different jump rates cancel in the limit. The results are proved by first finding an auxiliary stochastic ranking process, for which a strong law of large numbers is applied, and then applying a multi time recursive Gronwall's inequality. The limit has a representation in terms of non-Markovian processes which we call point processes with last-arrival-time dependent intensities. We also prove the propagation of chaos, i.e., the tagged particle processes also converge almost surely.

Mathematics Subject Classification. Primary 60K35; Secondary 82C22.
Key words and phrases. Stochastic ranking process, hydrodynamic limit, law of large numbers, complete convergence, last-arrival-time dependent intensity, Gronwall inequality.

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