MALLIAVIN CALCULUS ON THE CLIFFORD ALGEBRA

We deal with Malliavin calculus on the L2 space of the W∗algebra generated by fermion fields (the Clifford algebra). First, we verify the product formula for multiple integrals in Itˆo-Clifford calculus, which is Itˆo calculus on the Clifford algebra. Using this product formula, we can define the derivation operator and the divergence operator. The derivation operator and the divergence operator satisfy the canonical anti-commutation relations, and the divergence operator serves as an extension of the Itˆo-Clifford stochastic integral, satisfying the Clark-Ocone formula. Subsequently, using this calculus, we consider the concentration inequality, the logarithmic Sobolev inequality, and the fourth-moment theorem. Among these, the results concerning the fourth-moment theorem imply that, unlike in the case of the usual Brownian motion, convergence in distribution cannot be deduced from the convergence of the fourth moment.