On instability of global path properties of symmetric Dirichlet forms under Mosco-convergence

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