Spectral analysis of the multi-dimensional diffusion operator with random jumps from the boundary

TL;DR

This paper develops a Hilbert-space framework for analyzing a diffusion process with boundary jumps, characterizing its spectral properties, and providing conditions for the reality of its eigenvalues, advancing understanding of boundary-driven stochastic systems.

Contribution

It introduces a novel Hilbert-space approach to the diffusion process with boundary jumps, characterizes the generator's spectral properties, and proves the completeness of root vectors under certain conditions.

Findings

The spectrum is contained in a half-plane, but the numerical range covers the entire complex plane.

The system of root vectors is complete for absolutely continuous measures with square-integrable densities.

Conditions are provided for the non-real spectrum and the reality of the eigenvalue with smallest real part.

Abstract

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 introduced 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 characterise 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 non-zero 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.