  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.
