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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 8278161, 7 pages
https://doi.org/10.1155/2017/8278161
Research Article

Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

Correspondence should be addressed to Caishi Wang

Received 30 September 2016; Accepted 17 January 2017; Published 8 February 2017

Academic Editor: Antonio Scarfone

Copyright © 2017 Caishi Wang and Beiping Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 be a nonnegative function on . By using the Bernoulli annihilators, we first define in a dense subspace of -space of Bernoulli functionals a positive, symmetric, bilinear form associated with . And then we prove that is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with on -space of Bernoulli functionals, which we call the -Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, we show that the -Ornstein-Uhlenbeck semigroup is a Markov semigroup.