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

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

$$L_{\gamma,\mu}u\equiv -\frac12\Delta u+\gamma V_\mu(u),$$

with the Dirichlet boundary condition, where $V_\mu$ is a nonlocal ``$\mu$-centering'' potential defined by

$$V_\mu(u)=u-\int_Du d\mu.$$

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 $\gamma\gg1$ and $\gamma\ll1$. We also consider conditions on $\mu$ that guarantee that the principal eigenvalue is monotone increasing or decreasing in $\gamma$.