Prépublication de l'Institut Fourier no

by Stéphane ATTAL Prépublication de l'Institut Fourier no 495 (2000) ?fi fl?fi ? !fi?_$? fi fi? ?_$fl%& fl ' ( $ Abstract. — We show how the toy Fock space can be embedded into the usual Fock space of quantum stochastic calculus. This embedding gives rise to a rigorous discrete approximation of the Fock space and its natural noise operators. We recover the quantum Ito table from the discrete one. We finally show that the quantum Brownian motion and Poisson process can be simultaneously approached by quantum Bernoulli random walks. I. The toy Fock space. Let us realise a Bernoulli random walk on its canonical space. Let? ) *0, 1+, and - be the?-field generated by finite cylinders. One denotes by ?n the coordinate mapping : ?n .?/ ) ?n, for all n01 . Let p 0 20, 1 3 and q ) 14p. Let µp be the probability measure on .?, - / which makes the sequence . ?n /n5 , to be a sequence of independent, identically distributed Bernoulli random variables with law p?1 6 q?0. Let 7 p 3 8 2 denote the expectation with respect to µp . We have 7 p 3?n 2 ) 7 p 3?2n 2 ) p.

• fock space


• ?t


• bernoulli random


• thus ameasurable function


• space ?


• now define


• finite subsets


• quantum stochastic