# Spectral analysis for a generator of a discretizised jump diffusion

#### by Toralf Burghoff

Institution: Friedrich Schiller University Jena
Date: Thu 2nd May 2013
Time: 2:15 PM
Location: Old Physics-G16 (Jim Potter Room)

Abstract: We consider the infinitesimal generator of a Markov Chain $(Y_k^\varepsilon)_{k\ge0}$ induced by a discretization of the so called jump diffusion equation
$$X_t^\varepsilon(x)=x-\int_0^tU'(X_s^\varepsilon)ds+\varepsilon L_t$$
where $(L_t)_{t\ge0}$ is a symmetric $\alpha$−stable Lévy Process, $x\in R$ is the initial condition, $U$ is a smooth multi-well potential with local minima $m_1,\dots,m_n$ and $\varepsilon>0$ is a small control parameter.

The aim is to construct a Fourier-like expansion for this matrix with respect to the parameter $\varepsilon$, i.e. we try to investigate the behavior of eigenvalues and eigenvectors in the limit $\varepsilon\downarrow0$. First we show that, on a proper time scale that depends on $\varepsilon$, the chain reminds of another Markov Chain with state space $\{m_1, \dots, m_n\}$ and generator matrix $Q$. Then it is shown that there exists a $\varepsilon$-independent spectral gap such that the set of eigenvalues is separated in an interesting part that contains $n$ eigenvalues, one for each well of the potential $U$, and one that is of no interest for the expansion of the generator. Moreover, these eigenvalues coincide with the eigenvalues of $Q$ in the limit $\varepsilon\downarrow0$.

Finally, by using the concept of fundamental matrices we also establish a connection between the first hitting times of the wells and these aforementioned eigenvalues.