Stochastic analysis on extended sample space and a tightness results

Approximate Markov process defined on the extended sample space plays an important role in the theory of Dirichlet Space. In this paper, stochastic analysis is studied on the extended sample space including the Stratonovitch integral, Ito's formula, etc. and a tightness and continuity results about excursion laws of the processes associated to a sequence of Dirichlet spaces is obtained by following an idea of Lyons and Zheng. These results are applied back to the standard sample space, which improve a few results about the additive functionals and also enable us to obtain the Lyons and Zheng's tightness and continuity results in the situation where the processes may blow up and have killings. Connections between the Stratonovitch integrals on the extended sample space and that defined by Nakao with respect to a function in the Dirichlet space is made.