Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

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.