Spectral analysis of the multidimensional diffusion operator with random jumps from the boundary

0539466 - ÚJF 2022 RIV CH eng J - Článek v odborném periodiku
Krejčiřík, D. - Lotoreichik, Vladimir - Pankrashkin, K. - Tušek, M.
Spectral analysis of the multidimensional diffusion operator with random jumps from the boundary.
Journal of Evolution Equations. Roč. 21, č. 2 (2021), s. 1651-1675. ISSN 1424-3199. E-ISSN 1424-3202
Institucionální podpora: RVO:61389005
Klíčová slova: Brownian motion * Elliptic differential operator * non-self-adjoint and non-local boundary condition * numerical range * root vectors * spectral enclosures
Obor OECD: Pure mathematics
Impakt faktor: 1.261, rok: 2021
Způsob publikování: Omezený přístup
https://doi.org/10.1007/s00028-020-00647-1

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.
Trvalý link: http://hdl.handle.net/11104/0317203