SPECTRAL ANALYSIS OF A CLASS OF NONLOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH RANDOM JUMPS

Let D subset of R(d) be a bounded domain and let 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 it mu is an element of P(D). Prom 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 equivalent to -1/2 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 - integral(D) u 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 >> 1 and gamma << 1. We also consider conditions on mu that guarantee that the principal eigenvalue is monotone increasing or decreasing in gamma.