Feller semigroups for multidimensional diffusion processes (Q1363907)

The generator: \[ Au=\sum a_{ij}(x)u_{x_ix_j}(x)+\sum a_i(x)u_{x_i}(x)+a(x)u(x)- \int_{\overline Q}[u(x)-u(y)]m(x,dy) \] is considered on a bounded open domain Q in R\(^n\) with \(C^{\infty}\) boundary, \(C^{\infty}\) coefficients and nonlocal nontraversal boundary condition: \[ \gamma(x)u(x)+\int_{\overline Q}[u(x)-u(y)] \mu (x,dy)=0\quad (\forall x\in\partial Q). \] The case of traversal condition, where the term of \(\vec{\delta }(x)\nabla u(x)\) appears with \(\inf(|\vec{\delta}(x) |^2+\mu (x,\overline Q))\) \(>0\), was studied widely, for references, see the paper of \textit{K. Taira} [Mem. Am. Math. Soc. 99, No. 475, 65 p. (1992; Zbl 0794.47027)]. In the present paper, \(\inf(\gamma (x)+\mu (x,\overline Q))>0\) is supposed instead to meet the nontraversal situation. Under some mild conditions on the measure \(\nu (x,dy)\), the existence and uniqueness of a associated Feller semigroup is established.