## 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 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,μ}u =-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 ^{-1/2}λ _{0}(γ,μ)=∫ _{∂D}μ/ Vdσ2∫ _{D}1/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 _{k}γ ^{1-k/2}, for an explicit constant c _{k}.

Original language | English |
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Pages (from-to) | 1077-1093 |

Number of pages | 17 |

Journal | Communications in Contemporary Mathematics |

Volume | 13 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2011 |

## Keywords

- Brownian motion
- Principal eigenvalue
- non-local differential operator
- random space-dependent jumps

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics