  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.
