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Dynamics and self-similarity in min-driven clustering
Author(s):
Govind
Menon;
Barbara
Niethammer;
Robert
L.
Pego
Journal:
Trans. Amer. Math. Soc.
362
(2010),
6591-6618.
MSC (2010):
Primary 82C22
Posted:
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  is chosen with probability  , and the smallest cluster merges with  randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in  , 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:
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
Posted:
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 \textit{Analysis and Stochastics in Complex Physical Systems}.
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Copyright
2010,
American Mathematical Society
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