Sufficient conditions for the existence of Feller semigroups
The problem of the existence of nonnegative contraction semigroups for one-dimensional diffusion processes with nonlocal boundary conditions was considered by W. Feller. A. D. Ventsel' proved that if a second-order elliptic differential operator is the infinitesimal generator of a Feller semigroup, then its domain consists of functions satisfying nonlocal boundary conditions. Sufficient conditions for the elliptic operator with such a domain to be the infinitesimal generator of a Feller semigroup were studied in other papers. In these papers, it was assumed that the order of the nonlocal terms is lower than that of the boundary operator. The most difficult case in which the boundary and the nonlocal operators are of the same order was first considered by A. L. Skubachevskii under the condition that the nonlocal terms decrease near the boundary. We obtain sufficient conditions for the existence of Feller semigroups assuming that the condition of decreasing of the nonlocal terms may be violated near some subset of the boundary.
one-dimensional diffusion processes with nonlocal boundary conditions
second-order elliptic differential operator
infinitesimal generator of a Feller semigroup
scientific article; zbMATH DE number 1158912