Additive functionals of several time-reversible Markov processes

Abstract

I. 1. Additive functionals of Markov processes have been studied by many authors. We refer to [ 1 ] and [5] for the early history and to [ 161, I1719 [61, 181 f or more recent developments. The most complete results for the case of a time-reversible process are due to Fukushima [lo]. In this paper we study additive functionals of several independent time-reversible Markov processes. The results are useful, in particular, for investigating a certain class of Gaussian random fields which includes the free field of Euclidean quantum field theory and the so-called Brownian sheet [9]. Our approach is different from that of Fukushima. Even in the case of one Markov process his results do not imply ours. In the Introduction first, additive functionals of Brownian motions are discussed on an intuitive level. Then the general concept of standard timereversible Markov processes is outlined and the rigorous definition of additive functionals is given. On this base, the principal results of the paper are formulated. We mention also some open problems.