Introduction

This is the first paper in a series devoted to Green's and Dirichlet spaces. In the next publications we shall study the spaces associated with fine Markov processes and with a certain class of multiparameter processes.

For the Brownian motion with exponential killing, the Dirichlet space is Sobolev's space H1 and Green's space is the dual space H−1. Both spaces are widely used in the theory of the free field (arising in quantum field theory). General Dirichlet and Green's spaces can be applied in an analogous way to Gaussian random fields associated with Markov processes [2].

Axiomatic theory of Dirichlet spaces was developed by Beurling and Deny [1]. Silverstein [5] and Fukushima [3] investigated the relation between Dirichlet spaces and Markov processes.

We start from a symmetric Markov transition function and we deal simultaneously with a pair: the Dirichlet space H and Green's space K. They are in a natural duality and they play symmetric roles but, in some respects, K is simpler than H. We consider several models for K and H. In particular, we represent them by L-valued functions of time t where L is a functional Hilbert space. We get the conventional representation of H by passage to the limit as t → ∞. Analogously, letting t → 0, we arrive at a representation of K by distributions (generalized functions).