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首页> 外文期刊>Communications in Contemporary Mathematics >ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH SPATIALLY DEPENDENT RANDOM JUMPS
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ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH SPATIALLY DEPENDENT RANDOM JUMPS

机译：一类与空间相关的随机跳的布朗运动相关的非局部椭圆算子的本征值的渐近行为

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摘要

Let D ⊂ Rd be a bounded domain and let P(D) denote the space of probability measuresnon D. Consider a Brownian motion in D which is killed at the boundary and which,nwhile alive, jumps instantaneously according to a spatially dependent exponential clocknwith intensity γV to a new point, according to a distribution μ ∈ P(D). From its newnposition after the jump, the process repeats the above behavior independently of whatnhas transpired previously. The generator of this process is an extension of the operatorn−Lγ,μ, defined bynLγ,μu ≡ −n1n2n∆u + γV Cμ(u),nwith the Dirichlet boundary condition, where Cμ is the “μ-centering” operator defined bynCμ(u) = u − ZDnu dμ.nThe principal eigenvalue, λ0(γ, μ), of Lγ,μ governs the exponential rate of decay of thenprobability of not exiting D for large time. We study the asymptotic behavior of λ0(γ, μ)nas γ → ∞. In particular, if μ possesses a density in a neighborhood of the boundary,nwhich we call μ, thennlimnγ→∞nγ−12nλ0(γ, μ) = R∂Dnμ √Vndσn√2 RDn1nV dμn.nIf μ and all its derivatives up to order k−1 vanish on the boundary, but the kth derivativendoes not vanish identically on the boundary, then λ0(γ, μ) behaves asymptotically likenckγn1−kn2 , for an explicit constant ck.
机译：设D⊂Rd为有界域，设P（D）表示概率测度的空间D。考虑D中的布朗运动，该运动在边界处被杀死，并且在存在的同时根据强度随空间变化的指数时标瞬时跳跃。根据分布μ∈P（D）将γV移至新点。从跳转后的新位置开始，该过程将重复上述行为，而与先前发生的情况无关。此过程的生成器是由nLγ，μu定义的算子n-Lγ，μ的扩展，具有Dirichlet边界条件，其中Cμ是由nCμ定义的“μ中心”算子（u）= u-ZDnudμ.nLγ，μ的主要特征值λ0（γ，μ）决定了长时间不退出D的概率的指数衰减率。我们研究了λ0（γ，μ）nasγ→∞的渐近行为。特别是，如果μ在边界附近具有密度，我们称之为μ，则nnlimnγ→∞nγ-12nλ0（γ，μ）=R∂Dnμ√Vndσn√2RDn1nVdμn.nIfμ及其所有导数直到阶k-1在边界上消失，但第k个导数在边界上不完全消失，则对于一个明确的常数ck，λ0（γ，μ）渐近地表现为ckγn1-kn2。

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