Construction of a general class of Dirichlet forms in terms of white noise analysis (original) (raw)

1991, Stochastic Processes and their Applications

In the framework of white noise analysis a Gel'fand triple (Y) c (L2) = (W* has been defined (e.g. Kubo and Yokoi, 1989), the space of smooth test functionals (9) and the space of Hida distributions (L?)* play some important roles. It has been shown (e.g. Yokoi, 1990) that a positive Hida distribution @ is given by a positive measure v 0 on the space of real tempered distributions Y*. Thus the space (L')@ = L*(Y*; B , va) can be defined, where B is the Bore1 u-algebra on Y* generated by the weak topology. The present article is concerned with a special choice of pre-Dirichlet forms with domain (9) on (Lz)O which is a generalization of the energy form (Hida, Potthoff and Streit, 1988) and of the type > , for each FE (9) and where (H, A; j, ke No) is a double sequence of test functionals satisfying some natural conditions. Some closabihty results are given in the last section under mild conditions. Dirichlet forms * closable forms * white noise * test functionals * Hida distributions * directional derivative *second quantization F(U, v) = I (Vu, Vu), dp E