Coherent States in Bernoulli Noise Functionals

Abstract

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 l 2 (N) to L 2 ( , 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 ∈ l 2 (N) as the coherent states and prove that all the coherent states are total in L 2 ( , F, P). We also show that φ can be used to factorize L 2 ( , 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.

Wick Analysis for Bernoulli Noise Functionals

“…Note that the left-hand side of (57) can be strictly less than the right-hand side [11]. Using Proposition 30, we can get a growth estimate of the -transform of a generalized functional.…”

“…which belongs to 2 (Ω) (see [11] for details). It can be shown that ∈ (Ω) whenever ∈ S + (N) and moreover { | ∈ S + (N)} is total in (Ω).…”

“…Recently Wang et al [10] have presented an alternative approach to Privault's discretetime chaotic calculus. There are other works devoted to development of a theory of quantum Bernoulli noises (see, for instance, [11,12]).…”