The Annals of Probability

Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems

Peter H. Baxendale and Levon Goukasian

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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

Article information

Source
Ann. Probab., Volume 30, Number 1 (2002), 101-134.

Dates
First available in Project Euclid: 29 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1020107762

Digital Object Identifier
doi:10.1214/aop/1020107762

Mathematical Reviews number (MathSciNet)
MR1894102

Zentralblatt MATH identifier
1037.60053

Subjects
Primary: 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx] 37H20: Bifurcation theory [See also 37Gxx] 60J60: Diffusion processes [See also 58J65]

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

Citation

Baxendale, Peter H.; Goukasian, Levon. Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems. Ann. Probab. 30 (2002), no. 1, 101--134. doi:10.1214/aop/1020107762. https://projecteuclid.org/euclid.aop/1020107762


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