A CHARACTERIZATION OF GIBBS MEASURES ON /$R \times W_{0,0})^{Z^{\nu}}$ VIA STOCHASTIC CALCULUS

We consider Gibbs measures on $(R \times W_{0,0})^{Z^\nu}, W_{0,0} = {\omega \in C[0,1] : \omega(0) = \omega(1)}$, which are associated to an interaction between particles in lattice boson systems (quantum unbounded spin systems). In [4], the Gibbs measures were introduced in the study of equilibrium states of interacting lattice boson systems and were characterized by means of the equilibrium conditions. In this paper we utilize the techniques of the stochastic calculus of variations and the infinite dimensional Ito integral to derive stochastic equations which we call the equilibrium equations. We show that under appropriate conditions the equilibrium conditions and the equilibrium equations are equivalent. The lattice boson systems with superstable and regular interactions, which we studied in [4], are typical examples.