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.
- Brownian motion
- Principal eigenvalue
- non-local differential operator
- random space-dependent jumps
All Science Journal Classification (ASJC) codes
- Applied Mathematics