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Extra info for Recent Developments in Infinite-Dimensional Analysis and Quantum Probability: Papers in Honour of Takeyuki Hida’s 70th Birthday

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B7,Q x with r E N, i j E N, i j =1= ii , j =1= k. Thus the assertion follows . 8. The set {

By assumption (ii). Consequently, f-L E g,~~'l' (ii) =} (iii): Assume that (ii) holds. Let G E D(d l' ) be bounded and 8;ax(G, G) = 0. Since dill = 0, replacing G by G-essinf G, we may assume that G ~ 0 and f G df-L = 1. rCOO(M) . Let us introduce the measure v = G . f-L. Then ! F(x)f3~(x) ! -! rCOO(M) . This implies that v E g,~~ and by assumption (ii) G == 1. (iii) =} (ii): Assume that (iii) holds . Let v E g,~~, v = P . u. for a bounded function pEL I (M, f-L). Then °= ! f3~(x) dv(x) = ! f3~(x)p(x) df-L(x) STOCHASTIC DYNAMICS OF COMPA CT SPINS = - f id" p(x) , X(x» dJ1,(x) for all X E J7C OO(M p = const.

47). 7. 47) may be regarded as a general form unifying arithmetical formulas known under the name of Euler products [l, Chapter 11]. 3 below. 39) (N, mEN). 8 . Let t > s. Then, for all mEN and zED. 52) z ZB(t, z: A). (etEm(A) - z). J '""' cPm; A) = zy(N)n(N m) ", FA(N)t . 52) follows. 39). 53) with 0 ~ rj ~ aj, j = I, ... , n . 54) where C N,m > 0 is a normalization constant. For mEN and N E N, we mean by miN that m is a divisor of N. The set {\II N im }mlN of vectors is orthonormal. }. l We set :Fi ) := {aQJ( I a E C}.

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