2014/05/30 by Holcman, David, Schuss, Zeev
#34E20 35P20 60H10 60J60 60H30 60G40 35J25 82C31 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1405.7821
We demonstrate the oscillatory decay of the survival probability of the stochastic dynamics d\x_\eps=\mba(\x_\eps) dt +√(2\eps) \mbb(\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 \mba(\x). The boundary \p D of the domain is an unstable limit cycle of \mba(\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∑i,j=12 \frac\p 2[ σi,j(\x) u(\x) ]\p xi\p xj-∑i=12\frac \p [ ai(\x) u(\x)] \p xi =-λ_\eps u(\x), with \mbσ(\x)=\mbb(\x)\mbbT(\x). We calculate the leading-order asymptotic expansion of all eigenvalues λ_\eps for small \eps. The principal eigenvalue is known to decay exponentially fast as \eps→0. We find that for small \eps the higher-order eigenvalues are given by λm,n=nω1+miω2+O(\eps) for n=1,2,…, m=±1,…, where ω1 and ω2 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.