Fermion Ito's Formula II: The Gauge Process in Fermion Fock Space

Abstract

The stochastic calculus constructed in [2] for fermion Brownian motion is augmented through the inclusion of stochastic integration with respect to the gauge process. The solutions of certain non-commutative stochastic differential equations are used to construct dilations of contraction semigroups on a Hilbert space ${\mathfrak{h}}_0$ and of uniformly continuous, completely positive semigroups on $\mathbf{B}({\mathfrak{h}}_0)$. Finally we construct a fermion analogue of the classical Poisson process and investigate some of its properties.