Instability of a nonlinear oscillator with small friction and small additive noise

Abstract

Let $\lambda =\lambda (\beta ,\sigma ,a,b)$ denote the top Lyapunov exponent for the linearization along trajectories of the noisy damped non-linear oscillator $\ddot{x}+\beta \dot{x}+ax+b{x}^3=\sigma {\dot{W}}_t$, where $a$, $b$ and $\beta$ are all positive and $\sigma \ne 0$. In 2004 Arnold, Imkeller and Sri Namachchivaya stated without proof that $\lambda ({\epsilon }^2\beta ,\epsilon \sigma ,a,b)\sim \overline{\lambda }{\epsilon }^{2/3}$ as $\epsilon \to 0$ with $\overline{\lambda }>0$. This paper contains a proof of this assertion.

35 pages