Weakly integrable semigroups on locally convex spaces
type: Item
instance of: scholarly article
title: Weakly integrable semigroups on locally convex spaces
zbMATH Open document ID: 0589.47043
DOI: 10.1016/0022-1236(86)90063-7
author: Brian Jefferies
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: The standard theory of semigroups of continuous linear operators on a Banach space is not always well adapted to the study of Markov processes. For example, if the generator of a multidimensional diffusion process has discontinuous coefficients, then it is best viewed as acting in the space of bounded Borel measurable functions, with the smooth functions of compact support as its domain. However, such an operator cannot be expected to generate a strongly continuous semigroup in the usual sense. The present paper desribes a class of semigroups which treats diffusion semigroups, and is sufficiently general to apply to other examples, such as semigroups of unbounded operators on a Banach space. The semigroups are assumed to act on a locally convex space, and the basic property is that the resolvent should exist as a weak integral. The ''generator'' of the semigroup is then defined via the resolvent. The methods used exploit the relationship between the resolvent, and an associated integral equation. Further developements appear in a forthcoming paper ''The generation of weakly integrable semigroups'', J. Funct. Anal., in press.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 3946980
zbMATH Keywords: Markov processes
zbMATH Keywords: diffusion semigroups
zbMATH Keywords: generator
zbMATH Keywords: resolvent
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90063-7
OpenAlex ID: W2092549919
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description: scientific article; zbMATH DE number 3946980