TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 9, September 2009, Pages 5041–5060 S 0002-9947(09)04880-6 Article electronically published on April 16, 2009 SPECTRAL ANALYSIS OF A CLASS OF NONLOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH RANDOM JUMPS ROSS G. PINSKY Abstract. Let D ⊂ 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 γ> 0 to a new point, according to a distribution µ ∈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 γ,µ , defined by L γ,µ u ≡− 1 2 ∆u + γV µ (u), with the Dirichlet boundary condition, where V µ is a nonlocal “µ-centering” potential defined by V µ (u)= u −  D u dµ. The operator L γ,µ is symmetric only in the case that µ 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 γ,µ 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 γ  1 and γ  1. We also consider conditions on µ that guarantee that the principal eigenvalue is monotone increasing or decreasing in γ. 1. Introduction and statement of results Let D ⊂ R d be a bounded domain with C 2,α -boundary and let P (D) denote the space of probability measures on D. Fix a measure µ ∈P (D), and 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 γ> 0 to a new point, according to the distribution µ. From this new point it repeats the above behavior independently of what has transpired previously. Denote this process by X(t), and let τ D denote its lifetime. Denote probabilities and expectations for the Markov process X(t) starting from x ∈ D by P γ,µ x and E γ,µ x . Received by the editors June 18, 2007 and, in revised form, June 3, 2008. 2000 Mathematics Subject Classification. Primary 35P15, 60F10, 60J65. Key words and phrases. Principal eigenvalue, spectral analysis, Brownian motion, random jumps. This research was supported by the M. & M. Bank Mathematics Research Fund. c 2009 American Mathematical Society Reverts to public domain 28 years from publication 5041 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use