  We introduce a stochastic analysis of Grassmann random variables suitable for
the stochastic quantization of Euclidean fermionic quantum field theories.
Analysis on Grassmann algebras is developed here from the point of view of
quantum probability: a Grassmann random variable is an homomorphism of an
abstract Grassmann algebra into a quantum probability space, i.e. a
$C^{\ast}$-algebra endowed with a suitable state. We define the notion of
Gaussian processes, Brownian motion and stochastic (partial) differential
equations taking values in Grassmann algebras. We use them to study the long
time behavior of finite and infinite dimensional Langevin Grassmann stochastic
differential equations driven by Gaussian space-time white noise and to
describe their invariant measures. As an application we give a proof of the
stochastic quantization and of the removal of the space cut-off for the
Euclidean Yukawa model.
