Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Concentration inequalities for a removal-driven thinning process

Authors: Joe Klobusicky and Govind Menon
Journal: Quart. Appl. Math.
MSC (2010): Primary 35R60, 60K25, 82C23, 82C70
DOI: https://doi.org/10.1090/qam/1474
Published electronically: June 5, 2017
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Abstract: We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening introduced by the first author (2014). The system consists of $ n$ particles in $ (0,\infty )$ that move at unit speed to the left. Each time a particle hits the boundary point 0, it is removed from the system along with a second particle chosen uniformly from the particles in $ (0,\infty )$. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density $ f_0(x) \in L^1_+(0,\infty )$, the empirical measure of the particle system at time $ t$ is shown to converge to the measure with density $ f(x,t)$, where $ f$ is the unique solution to the kinetic equation with nonlinear boundary coupling

$\displaystyle \partial _t f (x,t) - \partial _x f(x,t) = -\frac {f(0,t)}{\int _0^\infty f(y,t)\, dy} f(x,t), \quad 0<x < \infty ,$    

and initial condition $ f(x,0)=f_0(x)$.

The proof relies on a concentration inequality for an urn model studied by Pittel, and Maurey's concentration inequality for Lipschitz functions on the permutation group.

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Additional Information

Joe Klobusicky
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180
Email: klobuj@rpi.edu

Govind Menon
Affiliation: Division of Applied Mathematics, Box F, Brown University, Providence, Rhode Island 02912
Email: govind_menon@brown.edu

DOI: https://doi.org/10.1090/qam/1474
Keywords: Piecewise-deterministic Markov process, functional law of large numbers, diminishing urns
Received by editor(s): April 27, 2017
Published electronically: June 5, 2017
Article copyright: © Copyright 2017 Brown University

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