On sub-Markov resolvents. The restriction to an open set and the Dirichlet problem
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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.
KeywordsFunction Versus Dirichlet Problem Convex Cone Vector Lattice Compact Space
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