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Transactions of the American Mathematical Society

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Stochastic averaging with a flattened Hamiltonian: A Markov process on a stratified space (a whiskered sphere)


Author: Richard B. Sowers
Journal: Trans. Amer. Math. Soc. 354 (2002), 853-900
MSC (1991): Primary 60F17; Secondary 37J40, 58A35, 60J35
DOI: https://doi.org/10.1090/S0002-9947-01-02903-8
Published electronically: October 4, 2001
MathSciNet review: 1867364
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Abstract: We consider a random perturbation of a 2-dimensional Hamiltonian ODE. Under an appropriate change of time, we identify a reduced model, which in some aspects is similar to a stochastically averaged model. The novelty of our problem is that the set of critical points of the Hamiltonian has an interior. Thus we can stochastically average outside this set of critical points, but inside we can make no model reduction. The result is a Markov process on a stratified space which looks like a whiskered sphere (i.e, a 2-dimensional sphere with a line attached). At the junction of the sphere and the line, glueing conditions identify the behavior of the Markov process.


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Additional Information

Richard B. Sowers
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
Email: r-sowers@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02903-8
Keywords: Markov processes, stochastic averaging, stratified space
Received by editor(s): September 7, 2000
Received by editor(s) in revised form: June 1, 2001
Published electronically: October 4, 2001
Additional Notes: This work was supported by NSF DMS 9615877 and NSF DMS 0071484. The author would also like to thank Professor Sri Namachchivaya of the Department of Aeronautical and Astronautical Engineering at the University of Illinois at Urbana-Champaign for the seemingly infinite time he donated to discussing the contents of this paper and without whose interest this subject would not have been considered. The author would also like to thank Professor Eugene Lerman of the Department of Mathematics at the University of Illinois at Urbana-Champaign for several helpful discussions about stratified spaces.
Article copyright: © Copyright 2001 American Mathematical Society