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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dynamics and self-similarity in min-driven clustering


Authors: Govind Menon, Barbara Niethammer and Robert L. Pego
Journal: Trans. Amer. Math. Soc. 362 (2010), 6591-6618
MSC (2010): Primary 82C22
Published electronically: July 20, 2010
MathSciNet review: 2678987
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Abstract: We study a mean-field model for a clustering process that may be described informally as follows. At each step a random integer $ k$ is chosen with probability $ p_k$, and the smallest cluster merges with $ k$ randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in $ (0,\infty)$, and we establish necessary and sufficient conditions for the approach to self-similar form. We also characterize eternal solutions for this model via a Lévy-Khintchine formula. The analysis is based on an explicit solution formula discovered by Gallay and Mielke, extended using a careful choice of time scale.


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Additional Information

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

Barbara Niethammer
Affiliation: Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom
Email: niethammer@maths.ox.ac.uk

Robert L. Pego
Affiliation: Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: rpego@cmu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05085-8
PII: S 0002-9947(2010)05085-8
Received by editor(s): July 28, 2008
Received by editor(s) in revised form: April 15, 2009
Published electronically: July 20, 2010
Additional Notes: This material is based upon work supported by the National Science Foundation under grant nos. DMS 06-04420, DMS 06-05006, DMS 07-48482, and by the Center for Nonlinear Analysis under NSF grants DMS 04-05343 and 06-35983
The second and third authors thank the DFG for partial support through a Mercator professorship for RLP at Humboldt University and through the Research Group Analysis and Stochastics in Complex Physical Systems.
Article copyright: © Copyright 2010 American Mathematical Society