Stochastic ranking process with space-time dependent intensities

TL;DR

This paper models the evolution of rankings with space-time dependent jump rates, proving convergence of empirical distributions and tagged particles to deterministic limits described by Burgers-like PDEs.

Contribution

It introduces a stochastic ranking process with space-time dependent intensities and characterizes its limit behavior via PDEs, extending previous models with new analytical results.

Findings

Empirical distribution converges almost surely to a deterministic limit.

Tagged particle processes also converge almost surely.

Limit distribution described by Burgers-like PDEs with evaporation.

Abstract

We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson times of jumps to the origin.