Spectral Analysis of a Discrete Metastable System Driven by L\'evy Flights

In this paper we consider a finite state time discrete Markov chain that mimics the behaviour of solutions of the stochastic differential equation $dX=-U^{\mathrm{\prime }}(X)dt+ϵdL$, where $U$ is a multi-well potential with $n\ge 2$ local minima and L is a symmetric \alpha-stable L'evy process (L'evy flights process). We investigate the spectrum of the generator of this Markov chain in the limit $ϵ\to 0$ and localize the top n eigenvalues ${\lambda }_1^{ϵ},\dots ,{\lambda }_n^{ϵ}$. These eigenvalues turn out to be of the same algebraic order $O({ϵ}^{\alpha })$ and are well separated from the rest of the spectrum by a spectral gap. We also determine the limits ${\lim }_{ϵ\to 0}{ϵ}^{-\alpha }{\lambda }_i^{ϵ}$, $1\le i\le n$, and show that the corresponding eigenvectors are approximately constant over the domains which correspond to the potential wells of $U$.