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

Abstract

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