## Tohoku Mathematical Journal

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

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

#### 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