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Analytic Feller semigroups. (English) Zbl 1074.47509

The author studies the existence and uniqueness of solutions of the evolution equation \(Wu=f\) on \(D\) and \(Lu=\varphi\) on \(\partial D\), where \[ Wu(x)=\sum_{i,j=1}^N a^{ij}(x){{\partial^2 u}\over {\partial x_i \partial x_j}}(x)+\sum_{i=1}^N b^i(x){{\partial u}\over {\partial x_i}}(x) +c(x)u(x) \]
\[ +\int_{{\mathbb{R}}^N\backslash \{0\}}\left( u(x+z)-u(x)-\sum^N_{j=1} z_j {{\partial u}\over {\partial x_j}}(x)\right)s(x,z)m(dz), \] \(D\) is a convex domain with \(C^\infty\)-boundary \(\partial D\), \(s(x,z)=0\) if \(x+z\not\in \overline{D}\), \(W\) is considered as an operator acting on functions on \(\overline{D}\), and the operator \(L\) is defined as \[ Lu(x')=\mu(x'){{\partial u}\over {\partial n}}(x')+\gamma(x')u(x') \] on \(\partial D\). He shows the existence of a Feller semigroup associated with \((W,L)\) and its analyticity.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47D07 Markov semigroups and applications to diffusion processes
60J35 Transition functions, generators and resolvents
35J15 Second-order elliptic equations
60J50 Boundary theory for Markov processes
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