A monotone convergence theorem for strong Feller semigroups

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2023
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Budde, Christian
Dobrick, Alexander
Glück, Jochen
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Zusammenfassung

For an increasing sequence (Tn) of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps 1 to a continuous function. This is instrumental in the study of elliptic operators on Rd with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and to show that the semigroup is strongly Feller under very mild regularity assumptions on the coefficients. We also provide counterexamples to demonstrate that the assumptions in our main result are close to optimal.

Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
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Strong Feller property, Monotone convergence, Parabolic PDEs with unbounded coefficients, Transition semigroup, Non-strongly continuous semigroups
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ISO 690BUDDE, Christian, Alexander DOBRICK, Jochen GLÜCK, Markus KUNZE, 2023. A monotone convergence theorem for strong Feller semigroups. In: Annali di Matematica Pura ed Applicata. Springer. 2023, 202(4), pp. 1573-1589. ISSN 0373-3114. eISSN 1618-1891. Available under: doi: 10.1007/s10231-022-01293-9
BibTex
@article{Budde2023monot-59952,
  year={2023},
  doi={10.1007/s10231-022-01293-9},
  title={A monotone convergence theorem for strong Feller semigroups},
  number={4},
  volume={202},
  issn={0373-3114},
  journal={Annali di Matematica Pura ed Applicata},
  pages={1573--1589},
  author={Budde, Christian and Dobrick, Alexander and Glück, Jochen and Kunze, Markus}
}
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