Spectral Analysis for a Discrete Metastable System Driven by Lévy Flights

In this paper we consider a finite state time discrete Markov chain that mimic the behaviour of solutions of the stochastic differential equation $$\begin{aligned} X_{t}^{\varepsilon }(x)=x-\int _0^t U^{\prime }(X_{s}^{\varepsilon })\, \mathrm {d}s+\varepsilon L_{t}, \end{aligned}$$Xtε(x)=x-∫0tU′(Xsε)ds+εLt,where U is a multi-well potential with $$n\ge 2$$n≥2 local minima and $$L=(L_t)_{t\ge 0}$$L=(Lt)t≥0 is a symmetric $$\alpha $$α-stable Lévy process (Lévy flights process). We investigate the spectrum of the generator of this Markov chain in the limit $$\varepsilon \rightarrow 0$$ε→0 and localize the top n eigenvalues $$\lambda ^\varepsilon _1,\ldots ,\lambda ^\varepsilon _n$$λ1ε,…,λnε. These eigenvalues turn out to be of the same algebraic order $$\mathcal O(\varepsilon ^\alpha )$$O(εα) and are well separated from the rest of the spectrum by a spectral gap. We also determine the limits $$\lim _{\varepsilon \rightarrow 0}\varepsilon ^{-\alpha } \lambda ^\varepsilon _i$$limε→0ε-αλiε, $$1\le i\le n$$1≤i≤n, and show that the corresponding eigenvectors are approximately constant over the domains which correspond to the potential wells of U.