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

We demonstrate the oscillatory decay of the survival probability of the stochastic dynamics $d\x_\eps=\mb{a}(\x_\eps)\, dt +\sqrt{2\eps}\,\mb{b}(\x_\eps)\,d\w$, which is activated by small noise over the 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 perturbation expansion of the spectrum of the Dirichlet problem for the non-self adjoint Fokker-Planck operator in $D$ $$L_\eps u(\x)=\,\eps\sum_{i,j=1}^2 \frac{\p ^2\left[ \sigma ^{i,j}\left(\x\right) u(\x) \right]}{\p x^i\p x^j}-\sum_{i=1}^2\frac {\p \left[ a^i\left(\x\right) u(\x)\right]} {\p x^i} =-\lambda_\eps u(\x),$$ with $\mb{\sigma}(\x)=\mb{b}(\x)\mb{b}^T(\x)$. We calculate the leading-order asymptotic expansion of all eigenvalues $\lambda_\eps$ for small $\eps$. The principal eigenvalue is known to decay exponentially fast as $\eps\to0$. We find that for small $\eps$ the higher-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 explicitly computed constants. We also find the asymptotic structure of the eigenfunctions of $L_\eps$ and of its adjoint $L^*_\eps$. We illustrate the oscillatory decay with a model of synaptic depression of neuronal network in neurobiology.