Geodesic
On the Existence of a~Feller Semigroup with Atomic Measure in a~Nonlocal Boundary Condition
Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 164-179

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The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region $G$. Its domain of definition consists of continuous functions satisfying a nonlocal condition on the boundary of the region. In general, the nonlocal term is an integral of a function over the closure of the region $G$ with respect to a nonnegative Borel measure $\mu(y,d\eta)$, $y\in\partial G$. It is proved that the operator is a generator of a Feller semigroup in the case where the measure is atomic. The smallness of the measure is not assumed. 
@article{TRSPY_2008_260_a10,
     author = {P. L. Gurevich},
     title = {On the {Existence} of {a~Feller} {Semigroup} with {Atomic} {Measure} in {a~Nonlocal} {Boundary} {Condition}},
     journal = {Informatics and Automation},
     pages = {164--179},
     publisher = {mathdoc},
     volume = {260},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a10/}
}
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P. L. Gurevich. On the Existence of a~Feller Semigroup with Atomic Measure in a~Nonlocal Boundary Condition. Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 164-179. http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a10/