Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure
Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure
复制标题
DOI：
10.1016/j.jfa.2007.05.019
复制
发表时间：
2007-07
影响因子：
1.7
通讯作者：
Iddo Ben Ari;R. Pinsky
中科院分区：
文献类型：
--
作者：
Iddo Ben Ari;R. Pinsky
设D为有界域，D上的二阶椭圆算子为D上的二阶椭圆算子，ν为D上的概率测度，L表示其定义域为如下非局部边界条件的微分算子：且在定义域上与L重合.显然，0是L的本征值，对应的本征函数是常量。众所周知，L具有无穷多个本征值序列，且除零本征值外，所有本征值都有负实部。定义了由ν标度的L的谱隙。在这篇文章中，我们研究了L的一般本征值，特别是谱隙γ1(ν)。算符L及其谱隙γ1(ν)具有概率意义。算子L是一个从边界具有随机跳跃的扩散过程的生成元，γ1(ν)度量了这个过程对其不变测度的指数收敛速度。
Let D⊂Rdbe a bounded domain and let be a second-order elliptic operator on D. Let ν be a probability measure on D. Denote by L the differential operator whose domain is specified by the following nonlocal boundary condition: and which coincides with L on its domain. Clearly 0 is an eigenvalue for L, with the corresponding eigenfunction being constant. It is known that L possesses an infinite sequence of eigenvalues, and that with the exception of the zero eigenvalue, all eigenvalues have negative real part. Define the spectral gap of L, indexed by ν, by In this paper we investigate the eigenvalues of L in general and the spectral gap γ1(ν) in particular. The operator L and its spectral gap γ1(ν) have probabilistic significance. The operator L is the generator of a diffusion process with random jumps from the boundary, and γ1(ν) measures the exponential rate of convergence of this process to its invariant measure.