Coherent States in Bernoulli Noise Functionals

Let (, F, P) be a probability space and Z = (Z K )k∈N a Bernoulli noise on (, 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 φ from l2(N) to L2(, F, P) which is called the coherent mapping. We prove that φ has the continuity property and other properties of operation. We then define functionals of the form φ( f ) with f ∈ l2(N) as the coherent states and prove that all the coherent states are total in L2(, F, P). We also show that φ can be used to factorize L2(, F, P). Finally we give an application of the coherent states to calculus of quantum Bernoulli noise. 2010 Mathematics subject classification: primary 60H40; secondary 46E30.