Malliavin calculus on the Clifford algebra

Abstract

We deal with Malliavin calculus on the $L^2$ space of the $W^*$-algebra generated by fermion fields (the Clifford algebra). First, we verify the product formula for multiple integrals in It\^o-Clifford calculus, which is It\^o calculus on the Clifford algebra. Using this product formula, we can define the derivation operator and the divergence operator. Anti-symmetric Malliavin calculus thus constructed has properties similar to those of usual Malliavin calculus. The derivation operator and the divergence operator satisfy the canonical anti-commutation relations, and the divergence operator serves as an extension of the It\^o-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. As for the logarithmic Sobolev inequality, only a weaker result is obtained. On the other hand, as for the concentration inequality, we obtain results that are almost similar to those of the usual case; moreover, 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.