Spectral Analysis of a Discrete Metastable System Driven by Lévy Flights

Abstract

In this paper we consider a finite state time discrete Markov chain that mimics the behaviour of solutions of the stochastic differential equation $dX=-U^{\prime }(X)dt+\epsilon dL$, where $U$ is a multi-well potential with $n\ge 2$ 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 $\epsilon \to 0$ and localize the top n eigenvalues ${\lambda }_1^{\epsilon },\dots ,{\lambda }_n^{\epsilon }$. 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 ${\lim }_{\epsilon \to 0}{\epsilon }^{-\alpha }{\lambda }_i^{\epsilon }$, $1\le i\le n$, and show that the corresponding eigenvectors are approximately constant over the domains which correspond to the potential wells of $U$.

21 pages, 3 figures

References in corpus (1)

• Kramers' law: Validity, derivations and generalisations