By Persi Diaconis, David Elworthy, Hans Föllmer, Edward Nelson, George Papanicolaou, Srinivasa R.S. Varadhan, Paul-Louis Hennequin

This quantity comprises distinctive, worked-out notes of six major classes given on the Saint-Flour summer time colleges from 1985 to 1987.

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X I4 ) XI XI))^)@ > where a , b E R, the noise operators are defined on the same interval I and @(I) = p ( I ) . Using the notation < 2 >=< @,a:@ > this amounts to the positive semi-definiteness of the quadratic form A= [ << B;,(xI)Bi"(XI) >> << %,(XI) (B,n(XI))2 ( q ( X I ) ) 2 ( p ( x I ) ) 2 (pa(xI))2 1 > . c2"-'p(I) ( 2 n ) ! )2C2n-2p(1)2 ) I. + ((q! )2)C2n-3p(1) A is a symmetric matrix, so it is positive semi-definite if and only if its minors are non-negative. ; Thus the interval I cannot be arbitrarily small.

C2"-'p(I) ( 2 n ) ! )2C2n-2p(1)2 ) I. + ((q! )2)C2n-3p(1) A is a symmetric matrix, so it is positive semi-definite if and only if its minors are non-negative. ; Thus the interval I cannot be arbitrarily small. 0 3. The q-Deformed Fock Case In the q-deformed case, where q E (-1, l ) , q # 0, we start with the q-white noise commutation relations at at - q af;at = q t - s) and letting, as in the Boson case, B z ( f ) := JRd f ( t ) a L n @ d t we obtain the q-RPWN commutation relations where 31 = { ;(n-A)(k-A) [k[klq!

L ] . G. Cubillo is grateful to L. Accardi and Centro Vito Volterra for support and kind hospitality. References 1. L. G. Lu, I. Volovich, Quantum Theory and Its Stochastic Limit, Springer-Verlag, Berlin, 2002. 2. L. V. Kozyrev, Quantum Interacting Particle Systems. In Quantum Interacting Particle Systems, World Scientific, Singapore, 2002, pp. 1193. 3. M. E. Shilov, Les Distributions, Dunod, Paris, 1962. 4. G. Maz’ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. 5. L. Schwartz, Mkthodes Mathkmatiques pour les Sciences Physiques, Hermann, Paris, 1966.

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