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

IV Section — Potential Theory

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

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.

Keywords

  • Function Versus
  • Dirichlet Problem
  • Convex Cone
  • Vector Lattice
  • Compact Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliographie

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

    MATH  Google Scholar 

  2. Constantinescu, C., Cornea, A., Potential Theory on Harmonic Spaces, Springer, Berlin-Heidelberg-New-York, 1972.

    CrossRef  MATH  Google Scholar 

  3. Meyer, P.A., Processus de Markov. Notes Math. 26, Springer, Berlin-Heidelberg-New York, 1967.

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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.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Steica, L., Local operators and Markov Processes, Lect.Notes Math. 816, Springer, Berlin-Heidelberg-New York, 1980.

    CrossRef  Google Scholar 

  7. Taylor, J.C., Ray processes on locally compact spaces, Math.Ann. 208, 233–248, 1974.

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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

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