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 dxε = a(xε) dt+ √ 2ε b(xε) dw, which is activated by small noise over the boundary of the domain of attraction D of a stable focus of the drift a(x). The boundary ∂D of the domain is an unstable limit cycle of 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 inD Lεu(x) = ε ∑2 i,j=1 ∂[σ(x)u(x)] ∂xi∂xj −∑2i=1 ∂[a (x)u(x)] ∂xi = −λεu(x), with σ(x) = b(x)bT (x). We calculate the leading-order asymptotic expansion of all eigenvalues λε for small ε. The principal eigenvalue is known to decay exponentially fast as ε → 0. We find that for small ε the higher-order eigenvalues are given by λm,n = nω1+miω2+O(ε) for n = 1, 2, . . . , m = ±1, . . . , where ω1 and ω2 are explicitly computed constants. We also find the asymptotic structure of the eigenfunctions of Lε and of its adjoint L∗ε . We illustrate the oscillatory decay with a model of synaptic depression of neuronal networks in neurobiology.