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Unbounded perturbations of two-dimensional diffusion processes with nonlocal boundary conditions. (English. Russian original) Zbl 1155.47049

Dokl. Math. 76, No. 3, 891-895 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 417, No. 4, 451-455 (2007).
Let \(G\subset \mathbb R^2\) be a bounded domain and \(K\) a finite subset of the boundary \(\partial G\). Let \(\Gamma_i\) be open connected such that \(\partial G\setminus K= \bigcup_{i=1}^N \Gamma_i\). This paper investigates perturbations of a uniform elliptic \(P_0\) differential operator with nonlocal boundary conditions like \(u- B_i u +\psi_i\) on \(\Gamma_i\), \(1\leq i\leq N.\) Under certain regularity conditions on \(P_0\) and a perturbation operator \(P_1\), it is proved that \(P_0+P_1\) has a proper closure which generates a Feller semigroup. Since \(P_0\) is uniform elliptic, it should be reasonable to ask further for the strong Feller property.

MSC:

47F05 General theory of partial differential operators
60J60 Diffusion processes
47D06 One-parameter semigroups and linear evolution equations
47A55 Perturbation theory of linear operators
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References:

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