COHERENT STATES IN BERNOULLI NOISE FUNCTIONALS

Let (Omega, F, P) be a probability space and Z = (ZK)(k is an element of N) a Bernoulli noise on (Omega, F, P) which has the chaotic representation property. In this paper, we investigate a special family of functionals of Z, which we call the coherent states. First, with the help of Z, we construct a mapping phi from l(2)(N) to L(2)(Omega, F, P) which is called the coherent mapping. We prove that phi has the continuity property and other properties of operation. We then define functionals of the form phi(f) with f is an element of l(2)(N) as the coherent states and prove that all the coherent states are total in L(2)(Omega, F, P). We also show that phi can be used to factorize L(2)(Omega, F, P). Finally we give an application of the coherent states to calculus of quantum Bernoulli noise.