Representation of the generator and the boundary condition for semigroups of operators of kernel type
type: Item
instance of: scholarly article
title: Representation of the generator and the boundary condition for semigroups of operators of kernel type
zbMATH Open document ID: 0552.47018
author: Tadashi Ueno
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 47Gxx
zbMATH DE Number: 3879572
zbMATH Keywords: representation of generator
zbMATH Keywords: general boundary condition
zbMATH Keywords: boundary condition of Markovian semigroups
zbMATH Keywords: second order elliptic integro- differential operators
zbMATH Keywords: regularity conditions
zbMATH Keywords: complex valued kernels
MaRDI profile type: Publication
full work available at URL: pjaa.59.414
OpenAlex ID: W2074942082
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DOI: 10.3792/PJAA.59.414
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description: scientific article; zbMATH DE number 3879572