Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

Abstract:

Let $D\subset R^d$ be a bounded domain and let $\mathcal P(D)$ denote the space of probability measures on $D$. Consider a Brownian motion in $D$ which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity $\gamma >0$ to a new point, according to a distribution $\mu \in \mathcal P(D)$. From this new point it repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator $-L_{\gamma ,\mu }$, defined by \begin{equation*} L_{\gamma ,\mu }u\equiv -\frac 12\Delta u+\gamma V_\mu (u), \end{equation*} with the Dirichlet boundary condition, where $V_\mu$ is a nonlocal “$\mu$-centering” potential defined by \begin{equation*} V_\mu (u)=u-\int _Du d\mu . \end{equation*} The operator $L_{\gamma ,\mu }$ is symmetric only in the case that $\mu$ is normalized Lebesgue measure; thus, only in that case can it be realized as a selfadjoint operator. The corresponding semigroup is compact, and thus the spectrum of $L_{\gamma ,\mu }$ consists exclusively of eigenvalues. As is well known, the principal eigenvalue gives the exponential rate of decay in $t$ of the probability of not exiting the domain by time $t$. We study the behavior of the eigenvalues, our main focus being on the behavior of the principal eigenvalue for the regimes $\gamma \gg 1$ and $\gamma \ll 1$. We also consider conditions on $\mu$ that guarantee that the principal eigenvalue is monotone increasing or decreasing in $\gamma$.