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Spectral analysis of the multidimensional diffusion operator with random jumps from the boundary
Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2021-01-04 , DOI: 10.1007/s00028-020-00647-1
David Krejčiřík , Vladimir Lotoreichik , Konstantin Pankrashkin , Matěj Tušek

We develop a Hilbert space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is defined by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterize the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the nonzero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.



中文翻译:

多维扩散算子从边界处随机跳跃的频谱分析

我们开发了一种希尔伯特空间方法,用于有界域中布朗运动的扩散过程,该运动具有自Ben-Ari和Pinsky在2007年引入的边界的随机跳跃。该过程的生成器由空间中的扩散椭圆型微分算子定义平方可积函数的服从非自伴和非局部边界条件,该条件通过域上的概率测度表示。我们获得了一个表达式,用于表示算符的解析度与其Dirichlet实现的解析度之间的差异。我们证明了数值范围是整个复杂平面,尽管频谱是纯离散的并且包含在半平面中。此外,对于一类具有平方可积密度的绝对连续概率测度,我们对伴随算子进行了刻画,并证明了根矢量的系统是完整的。最后,在密度的某些假设下,我们获得了非真实频谱的包围,并为具有最小真实部分的非零特征值找到了充分的条件。后者支持Ben-Ari和Pinsky的猜想,即该特征值始终是真实的。

更新日期:2021-01-04
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