# Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

## TL;DR

This paper constructs Dirichlet forms from Bernoulli annihilation operators, demonstrating their properties and linking them to a new class of Markov semigroups called the w-Ornstein-Uhlenbeck semigroup.

## Contribution

It introduces a novel method of constructing Dirichlet forms on Bernoulli functionals using annihilation operators, establishing their mathematical properties and associated Markov semigroups.

## Key findings

- The form $\\mathcal{E}_w$ is positive, symmetric, closed, and has the contraction property.
- The $w$-Ornstein-Uhlenbeck semigroup is shown to be a Markov semigroup.
- Dirichlet forms are successfully constructed from Bernoulli annihilators.

## Abstract

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anti-commutation 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 $\mathbb{N}$. By using the Bernoulli annihilators, we first define in a dense subspace of the $L^2$-space of Bernoulli functionals a positive, symmetric bilinear form $\mathcal{E}_w$ associated with $w$. And then we prove that $\mathcal{E}_w$ 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 the $L^2$-space of Bernoulli functionals, which we call the $w$-Ornstein-Uhlenbeck semigroup, and by using the Dirichlet form $\mathcal{E}_w$ we show that the $w$-Ornstein-Uhlenbeck semigroup is a Markov semigroup.