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

  • Lucreţiu Stoica
IV Section — Potential Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1014)


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


  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.


Function Versus Dirichlet Problem Convex Cone Vector Lattice Compact Space 
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Copyright information

© Springer-Verlag 1983

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  • Lucreţiu Stoica

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