Weakly integrable semigroups on locally convex spaces
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.
scientific article; zbMATH DE number 3946980
scientific article