Tohoku Mathematical Journal

Cancellation of fluctuation in stochastic ranking process with space-time dependent intensities

Tetsuya Hattori

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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.

Article information

Source
Tohoku Math. J. (2), Volume 71, Number 3 (2019), 359-396.

Dates
First available in Project Euclid: 18 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1568772177

Digital Object Identifier
doi:10.2748/tmj/1568772177

Mathematical Reviews number (MathSciNet)
MR4012354

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Keywords
stochastic ranking process hydrodynamic limit law of large numbers complete convergence last-arrival-time dependent intensity Gronwall inequality

Citation

Hattori, Tetsuya. Cancellation of fluctuation in stochastic ranking process with space-time dependent intensities. Tohoku Math. J. (2) 71 (2019), no. 3, 359--396. doi:10.2748/tmj/1568772177. https://projecteuclid.org/euclid.tmj/1568772177


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