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.
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The research was supported by the Czech-French MOBILITY project No. 8J18FR033. D.K. was supported by the GACR grants No. 18-08835S and 20-17749X of the Czech Science Foundation. K.P. was supported in part by the PHC Amadeus 37853TB funded by the French Ministry of Foreign Affairs and the French Ministry of Higher Education, Research and Innovation. M.T. was supported by the project CZ.02.1.01/0.0/0.0/16_019/0000778 from the European Regional Development Fund. The authors wish to express their thanks to Sergey Denisov for stimulating discussions. The authors are also grateful to the anonymous referee whose suggestions led to improvements in the manuscript.
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Krejčiřík, D., Lotoreichik, V., Pankrashkin, K. et al. Spectral analysis of the multidimensional diffusion operator with random jumps from the boundary. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028-020-00647-1
- Brownian motion
- Elliptic differential operator
- Non-self-adjoint and non-local boundary condition
- Numerical range
- Root vectors
- Spectral enclosures