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
Citation
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
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