Abstract
We give sufficient conditions for Mosco convergences for the following three cases: sym- metric locally uniformly elliptic diffusions, symmetric Levy processes, and symmetric jump processes in terms of the L 1 (R d ;dx)-local convergence of the (elliptic) coefficients, the charac- teristic exponents and the jump density functions, respectively. We stress that the global path properties of the corresponding Markov processes such as recurrence/transience, and conserva- tiveness/explosion are not preserved under Mosco convergences and we give several examples where such situations indeed happen.
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