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We demonstrate the oscillatory decay of the survival probability of thestochastic dynamics $d\x_\eps=\mb{a}(\x_\eps)\, dt+\sqrt{2\eps}\,\mb{b}(\x_\eps)\,d\w$, which is activated by small noise overthe boundary of the domain of attraction $D$ of a stable focus of the drift$\mb{a}(\x)$. The boundary $\p D$ of the domain is an unstable limit cycle of$\mb{a}(\x)$. The oscillations are explained by a singular perturbationexpansion of the spectrum of the Dirichlet problem for the non-self adjointFokker-Planck operator in $D$ \[L_\eps u(\x)=\,\eps\sum_{i,j=1}^2 \frac{\p^2\left[ \sigma ^{i,j}\left(\x\right) u(\x) \right]}{\p x^i\px^j}-\sum_{i=1}^2\frac {\p \left[ a^i\left(\x\right) u(\x)\right]} {\p x^i}=-\lambda_\eps u(\x),\] with $\mb{\sigma}(\x)=\mb{b}(\x)\mb{b}^T(\x)$. Wecalculate the leading-order asymptotic expansion of all eigenvalues$\lambda_\eps$ for small $\eps$. The principal eigenvalue is known to decayexponentially fast as $\eps\to0$. We find that for small $\eps$ thehigher-order eigenvalues are given by$\lambda_{m,n}=n\omega_1+mi\omega_2+O(\eps)$ for$n=1,2,\ldots,\,m=\pm1,\ldots$, where $\omega_1$ and $\omega_2$ are explicitlycomputed constants. We also find the asymptotic structure of the eigenfunctionsof $L_\eps$ and of its adjoint $L^*_\eps$. We illustrate the oscillatory decaywith a model of synaptic depression of neuronal network in neurobiology.

Oscillatory survival probability and eigenvalues of the non-self adjoint Fokker-Planck operator