Asymptotic behavior of the principal eigenvalue for a class of non-local elliptic operators related to brownian motion with spatially dependent random jumps

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 according to a spatially dependent exponential clock with intensity γV to a new point, according to a distribution μ ∈ P(D). From its new position after the jump, the process 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,μu =-1/2 Δu + γVCμ(u), with the Dirichlet boundary condition, where C _μ is the "μ-centering" operator defined by C _μ(u) = u-∫ _D u, dμ. The principal eigenvalue, λ ₀(γ, μ), of L _γ,μ governs the exponential rate of decay of the probability of not exiting D for large time. We study the asymptotic behavior of λ ₀(γ, μ) as γ → ∞. In particular, if μ possesses a density in a neighborhood of the boundary, which we call μ, then limγ→∞γ ^-1/2λ ₀(γ,μ)=∫ _∂Dμ/ Vdσ2∫ _D1/Vdμ. If μ and all its derivatives up to order k-1 vanish on the boundary, but the kth derivative does not vanish identically on the boundary, then λ ₀(γ, μ) behaves asymptotically like c _kγ ^1-k/2, for an explicit constant c _k.