A Problem From Hamiltonian Mechanics With Time -Periodic Coefficients, Small Noise, and Degeneracy
O'Bryant, Natella V.
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https://hdl.handle.net/2142/86804
Description
Title
A Problem From Hamiltonian Mechanics With Time -Periodic Coefficients, Small Noise, and Degeneracy
Author(s)
O'Bryant, Natella V.
Issue Date
2002
Doctoral Committee Chair(s)
Sowers, Richard B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
We consider a two-dimensional weakly dissipative dynamical system with time-periodic co-efficients. Their time average is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. The periodic fluctuations of the coefficients occur on the time scale of order 1/epsilon2, the effect of the drift is visible on the time scale of order 1/epsilon, and the diffusion coefficients are of order 1. Using the martingale-problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficient, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of the time scales, are similar to the conditions on the domain of skew Brownian motion, and are related to the description of spider martingales. The dynamics of the reduced process can also be understood through the Fokker-Planck equation, and the gluing coefficients arise in the corresponding conservation of flux and in the continuity equation.
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