(i) is proved similarly. It is worth mentioning at this point De Masi's result ((3]) that translation invariant DLR measures for the spin systems considered here are tempered.

I) They satisfy the We~l relation W(f)W(g) = exp{-i Im}W(f+g)(f,g ~L2[O,=)). 2) Introducing the mutually adjoint annihilation and creation o_onerators a(f), a+(f) by means of their actions on exponential vectors a(f)@(g) = @(g) a*(f)@(g) = d ~ ( g + t f ) t=O and noticing that a*(f) - a(f) is essentially skew-self-adjoint, we may write W(f) =--mxp{a*(f)-a(f)}, quantum Brownian motion [1,5,6] is the family of operators At : A t% a(X[O,t])' = The duality transformation (A t , A~, t ~ 0) a%(X[O,t]) [8] is a Hilbert space isomorphism, which we may use to identify the two spaces from H onto the Hilbert space L2(w), where w is Wiener measure, under conjugation by which the self-adjoint operators Qt = At + Aft' t -> O become multiplications by the canonical realisation X t, t z 0 of Brownian motion.

8. 19. W. Feller, An Introduction to Probability Theory and its Applications (two volumes)(Wiley and Sons, New York, 1966). A. Berezin and Ya. G. SJnai, Trudy Mosk. Mat. Obshch. ]7, 197-212 (1967). L. Dobrush~n, Proc. Fifth Berke]ey Symposium, III, 78-87 (1967). D. Ruelle, Statist~ea! B. Griffiths, Phys. Rev. 136, 437-439 (]964). M. T. V. Pu]~, J. Math. Anal. and App]. (~n press). M. E. Uhlenbeck and M. Kac, Physics Reports 32C, 169-248 1977). V. Pu1~, D. Ph~l. Thes~s, (Oxford, ]972). T. Cannon, Commun.