Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

Caishi Wang,Beiping Wang

doi:10.1155/2017/8278161

Abstract

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let w be a nonnegative function on N. By using the Bernoulli annihilators, we first define in a dense subspace of L2-space of Bernoulli functionals a positive, symmetric, bilinear form Ew associated with w. And then we prove that Ew is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with w on L2-space of Bernoulli functionals, which we call the w-Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, Ew we show that the w-Ornstein-Uhlenbeck semigroup is a Markov semigroup.

Highlights

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) admit much good operation properties with physical meanings
We prove that Ew interesting semigroup of operators is closed and has the associated with w on contraction L2-space of Bernoulli functionals, which we call the w-Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, Ew we show that the w-Ornstein-Uhlenbeck semigroup is a Markov semigroup
Privault [2] used the Bernoulli annihilators to define the gradients for Bernoulli functionals in 2008

Summary

Introduction

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) admit much good operation properties with physical meanings. They together with their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time [1]. [4], it has been shown that a wide class of quantum Markov semigroups can be constructed from the Bernoulli annihilators and their adjoint operators. In this paper, motivated by the work of Hida et al [12], we would like to apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Our work here does not mean that one can construct a Dirichlet form from quantum Bernoulli noises. Letters like j, k, and n stand for nonnegative integers, namely, elements of N

Bernoulli Annihilators

Forms Constructed from Bernoulli Annihilators

Contraction Property

Application

Discussion