  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.
