Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems

Baxendale, Peter H. ; Goukasian, Levon

Ann. Probab., Tome 30 (2002) no. 1, p. 101-134 / Harvested from Project Euclid

Résumé

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

Détail

Publié le : 2002-01-14
Classification:  Stochastic oscillator,  Lyapunov exponent,  Hamiltonian,  stochastic averaging,  nilpotent stochostic differential equation,  37H10,  60H10,  37H15,  37H20,  60J60