Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

Caishi Wang and Beiping Wang

Advances in Mathematical Physics, 2017, vol. 2017, issue 1

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.

Date: 2017
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https://doi.org/10.1155/2017/8278161

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2017:y:2017:i:1:n:8278161

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