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January 2002 Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems
Peter H. Baxendale, Levon Goukasian
Ann. Probab. 30(1): 101-134 (January 2002). DOI: 10.1214/aop/1020107762

Abstract

Consider the stochastic nonlinear oscillator equation

\[ \ddot{x} = -x -x^3 + \varepsilon^2 \beta \dot{x} + \varepsilon \sigma x \dot{W}_t \]

with $\beta < 0$ and $\sigma \neq 0$. If $4\beta + \sigma^2 > 0$ then for small enough $\varepsilon > 0$ the system $(x,\dot{x})$ is positive recurrent in ${\bf R}^2 \setminus \{(0,0\}$. Now let $\overline{\lambda}(\varepsilon)$ denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that

\[ \overline{\lambda}(\varepsilon) = \varepsilon^{2/3}\overline{\lambda} + O(\varepsilon^{4/3}) \qquad\mbox{as}\ \varepsilon \to 0 \]

with $\overline{\lambda} > 0$. This result depends crucially on the fact that the system above is a small perturbation of a Hamiltonian system. The method of proof can be applied to a more general class of small perturbations of two-dimensional Hamiltonian systems. The techniques used include (i) an extension of results of Pinsky and Wihstutz for perturbations of nilpotent linear systems, and (ii) a stochastic averaging argument involving

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Peter H. Baxendale. Levon Goukasian. "Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems." Ann. Probab. 30 (1) 101 - 134, January 2002. https://doi.org/10.1214/aop/1020107762

Information

Published: January 2002
First available in Project Euclid: 29 April 2002

zbMATH: 1037.60053
Digital Object Identifier: 10.1214/aop/1020107762

Subjects:
Primary: 37H10 , 60H10
Secondary: 37H15 , 37H20 , 60J60

Keywords: Hamiltonian , Lyapunov exponent , nilpotent stochostic differential equation , stochastic averaging , Stochastic oscillator

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 1 • January 2002
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