ADDENDUM TO THE PAPER DIRICHLET FORMS AND MARKOV PROCESSES A GENERALIZED FRAMEWORK INCLUDIN文档.pdfVIP

ADDENDUM TO THE PAPER DIRICHLET FORMS AND MARKOV PROCESSES A GENERALIZED FRAMEWORK INCLUDING BOTH ELLIPTIC AND PARABOLIC CASES Wilhelm Stannat Universit at Bielefeld Fakultat fur Mathematik Postfach Bielefeld Germany Abstract We extend the main result in AMR which is a complete characte rization of all Dirichlet forms dened on some L 2space L 2 E m asso ciated with mtight sp ecial standard pro cesses to the framework of generalized Dirichlet forms Introduction In our previous work St we intro duced the class of generalized Dirichlet forms which contains symmetric and co ercive Dirichlet forms as treated in Fu resp MR and time dep endent Dirichlet forms as considered in O but also examples of an entirely new kind such as fractional diusion op erators and htransformations cf St for a detailed discussion The main result in St is the identication of an analytic prop erty of generalized Dirichlet forms dened on L E m called quasi regularity in analogy with AMR and MR which allows the construction of an asso ciated mtight sp ecial standard pro cess This construction generalizes one half of the ab ove mentioned result in AMR namely the suciency part In the present pap er we also carry over the necessity part of the result in AMR to our more general situation More precisely let E b e a generalized semi Dirichlet form on L E m and assume that there exists an mtight sp ecial standard pro cess M asso ciated with E Then we shall prove in Theorem that E is quasiregular and moreover E is prop erly asso ciated in the resolvent sense with M cf Denition for the precise