Spectral Analysis of a Discrete Metastable System Driven by Lévy Flights

In this paper we consider a finite state time discrete Markov chain that mimics the behaviour of solutions of the stochastic differential equation $dX=-U'(X)dt+εdL$, where $U$ is a multi-well potential with $n\geq 2$ local minima and L is a symmetric α-stable Lévy process (Lévy 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 $λ^ε_1,\dots, λ^ε_n$. These eigenvalues turn out to be of the same algebraic order $O(ε^α)$ and are well separated from the rest of the spectrum by a spectral gap. We also determine the limits $\lim_{ε\to 0}ε^{-α} λ^ε_i$, $1\leq i\leq n$, and show that the corresponding eigenvectors are approximately constant over the domains which correspond to the potential wells of $U$.

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