The generation of weakly integrable semigroups
The semigroups of operators associated with Markov processes do not naturally fit into the scheme associated with semigroups strongly continuous at the origin. A diffusion process is best described by operators acting on the space of initial distributions of the diffusing substance. If arbitrary measures are allowed, then the resulting semigroup is rarely continuous in the variation norm. The present article deals with a class of semigroups of operators devised to deal with those arising from Markov processes, and systems of unbounded operators, introduced by the author in an earlier paper [J. Funct. Anal. 73, 195-215 (1987)]. A key feature of the approach is to appeal to the integrability properties of the semigroup, rather than differentiability, and so define the resolvent of the semigroup directly. The generator of the semigroup, when it exists, is then defined in terms of the resolvent. The problem addressed in this paper is to determine when a given resolvent family is the resolvent of a semigroup. A theorem of Widder showing how to recover a function from its Laplace transform is central to the argument here. The convergence of the Widder differential operators of the resolvent family is only in the weak topology. To ensure that the system converges in the space of operators, compactness arguments are used in place of the traditional appeal to completeness. The main result gives a set of conditions of the Hille-Yosida-Phillips type, which is both necessary and sufficient for a resolvent family to be the resolvent of a certain type of weakly integrable semigroup.
semigroups of operators associated with Markov processes
integrability properties of the semigroup
conditions of the Hille-Yosida-Phillips type
scientific article; zbMATH DE number 4007055