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 $dX=-U'(X)dt+\epsilon dL$, where $U$ is a multi-well potential with $n\geq 2$ 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 $\epsilon\to 0$ and localize the top n eigenvalues $\lambda^\epsilon_1,\dots, \lambda^\epsilon_n$. These eigenvalues turn out to be of the same algebraic order $O(\epsilon^\alpha)$ and are well separated from the rest of the spectrum by a spectral gap. We also determine the limits $\lim_{\epsilon\to 0}\epsilon^{-\alpha} \lambda^\epsilon_i$, $1\leq i\leq n$, and show that the corresponding eigenvectors are approximately constant over the domains which correspond to the potential wells of $U$.