Construction of a general class of Dirichlet forms in terms of white noise analysis

AbstractIn the framework of white noise analysis a Gel'fand triple (L)⊂(L2)⊂(L)∗ has been defined (e.g. Kubo and Yokoi, 1989), the space of smooth test functionals (L) and the space of Hida distributions (L)∗ play some important roles. It has been shown (e.g. Yokoi, 1990) that a positive Hida distribution Φ is given by a positive measure νΦ on the space of real tempered distributions L∗. Thus the space (L2)ΦL2(L∗; B, νΦ) can be defined, where B is the Borel σ-algebra on L∗ generated by the weak topology.The present article is concerned with a special choice of pre-Dirichlet forms with domain (L) on (L2)Φ which is a generalization of the energy form (Hida, Potthoff and Streit, 1988) and of the type ε(F)=Φ;∑j,kHj,k·∂jF·∂kF̄, for each Fϵ(L) and where (Hj,k; j, kϵN0) is a double sequence of test functionals satisfying some natural conditions. Some closability results are given in the last section under mild conditions