Convergence of finite dimensional distributions of heat kernel measures on loop groups

In this paper we consider heat kernel measure on loop groups associated to the H1/2-metric. Unlike Hs-case (s>1/2), there is a difficulty that H1/2 is not contained in the space of continuous loops. So we take limits. There are two limiting methods. One is to use delta functions and to let s go down to 1/2. The other is to fix s at 1/2 and to approximate the delta functions. For the second approach, a generalization of heat kernel measures is needed. Then, the first approach can be obtained as a special case of the second one. The limit in the sense of finite dimensional distribution is the fictitious infinite dimensional Haar measure.