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On sub-Markov resolvents. The restriction to an open set and the Dirichlet problem

  • IV Section — Potential Theory
  • Conference paper
  • First Online:
Complex Analysis — Fifth Romanian-Finnish Seminar

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1014))

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Abstract

This paper deals with sub-Markov resolvents (Vλ)λ>0 on a locally compact space E with countable base. The resolvent has some special properties the main of which are the following:

  1. 1o

    Vλ(Cc(E))⊂C(E),

  2. 2o

    There exists a standard process on Eassociated to the resolvent (Vλ).

For an open set U⊂E we study the resolvent (Vλ’)λ2>0 on U associated by killing the process on CU. Namely we give sufficient conditions (expressed by the existence of barrier functions) which imply that the resolvent (Vλ’)λ2>0 has properties of the type 1o (see Theorems 3.2 and 4.2). This problem is closely connected to the probabilistic Dirichlet problem (see Proposition 4.1 and Corollary 4.4).

Thanks are do to K.Janssen who made evident an error of the author.

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Bibliographie

  1. Blumenthal, R.M., Getoor, R.K., Markov Processes and Potential Theory, New York-London, Academic Press, 1968.

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  4. Mokobodzki, G., Cônes de potentiels et noyaux subordonés, in vol.Potential Theory, Edizione Cremonese, Roma, 1970.

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  5. Roth, J.P., Opérateurs dissipatifs et semi-groupes dans les espaces de fonctions continues, Ann.Inst.Fourier, 26, 1–98, 1976.

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Authors
  1. Lucreţiu Stoica
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Editor information

Cabiria Andreian Cazacu Nicu Boboc Martin Jurchescu Ion Suciu

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© 1983 Springer-Verlag

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Stoica, L. (1983). On sub-Markov resolvents. The restriction to an open set and the Dirichlet problem. In: Cazacu, C.A., Boboc, N., Jurchescu, M., Suciu, I. (eds) Complex Analysis — Fifth Romanian-Finnish Seminar. Lecture Notes in Mathematics, vol 1014. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072082

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  • DOI: https://doi.org/10.1007/BFb0072082

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12683-6

  • Online ISBN: 978-3-540-38672-8

  • eBook Packages: Springer Book Archive

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