Representation of the generator and the boundary condition for semigroups of operators of kernel type
The generator and the boundary condition of Markovian semigroups are represented by second order elliptic integro-differential operators under regularity conditions. Here, this fact is partially extended for complex valued kernels. Let Q(t,x,\(\cdot)\) be a complex valued bounded measure on a compact N- dimensional smooth manifold, and let \({\mathcal D}(A_{(x)})\) be the set of f in \(C_ b(D)\) such that \(Af(x)=\lim_{t\searrow 0}(Q_ tf(x)- f(x))/_ t\) exists. \(\{\xi_ k^{(x)}\), \(l\leq k\leq N\}\) denotes a local coordinate near x and is a set of smooth functions on D. If there is a number \(\ell\) such that \[ \int_{D}\sum_{l\leq k\leq N}| \xi_ k^{(x)}(y)|^{\ell}| Q| (t,x,dy)=0(t),\quad as\quad t\searrow 0, \] \[ \xi_{\alpha}^{(x)}=\xi^{(x)}_{\alpha_ l}...\xi^{(x)}_{\alpha_ N}\in {\mathcal D}(A_{(x)}),\quad \alpha =(\alpha_ l,\quad,\alpha_ N), \] then, for f in \({\mathcal D}(A_{(x)})\cap C^ n(D),\) \[ Af(x)=\sum_{| \alpha | \leq n}{\mathcal A}_{\alpha}(x)D_{\alpha}f(x)+\int_{D}(f(y)-\sum_{| \alpha | <m}(l/\alpha !)D_{\alpha}f(x)\xi_{\alpha}^{(x)}(y)))\mu (x,dy), \] where n is the smallest \(\ell\) above and \(\mu\) (x,\(\cdot)\) is a certain complex valued measure on \(D\setminus \{x\}\). In case D is a bounded open set in an N-dimensional manifold with smooth boundary \(\partial D\), a boundary condition of type \[ Lf(x)=\delta_{(x)}Af(x)+\sum_{\alpha \in I(n,n')}b_{\alpha}(x)D_{\alpha}f(x)+ \] \[ \int_{\bar D\setminus \{x\}}(f(y)-\sum_{\alpha \in I^ 0_{(n,n')}}(l/\alpha !)D_{\alpha}f(x)\xi_{\alpha}^{(x)}(y))\nu (t,dy)=0 \] is given for a boundary point x of D. Similar results for noncompact, or unbounded D, are given also.
boundary condition of Markovian semigroups
second order elliptic integro- differential operators
scientific article; zbMATH DE number 3879572
scientific article