Some Applications Of Quantum Bernoulli Noises

Quantum Bernoulli noises(QBN)are the family of annihilation and creation operators acting on Bernoulli functionals.Since they satisfy a canonical anti-commutation relation(CAR)in equal-time,they can characterize two-level quantum systems that have infinitely many quantum sites.In recent years,quantum Bernoul-li noises have been intensively studied and many deep results have been obtained of them.In this dissertation,we consider application of quantum Bernoulli nois-es to Dirichlet forms and nonlinear stochastic difference equations.The paper is organized as follows:In Chapter 1,we briefly describe the background and the state-of-art concern-ing our research topics.We also present some general concepts,and fundamental knowledge of quantum Bernoulli noises.In Chapter 2,we consider construction of Dirichlet forms by means of Bernoulli annihilation operators.Let ω be a nonnegative function on N.Firstly,by using the Bernoulli annihilation operators,we define a positive,symmetric bilinear form εωassociated with ω in a dense subspace of the L2-space of Bernoulli functionals.And then we prove that εω is closed and has the contraction property,hence it is a Dirich-let form.Finally,we consider an interesting semigroup of operators associated with w in the L2-space of Bernoulli functionals,which we call the ω-Ornstein-Uhlenbeck semigroup,and we show that the ω-Ornstein-Uhlenbeck semigroup is a Markov semigroup by using the Dirichlet form εω.Finally,in Chapter 3,we construct a class of nonlinear stochastic difference equations with the help of quantum Bernoulli noises,and examine the stability and boundedness of the solutions as well as other problems.