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

Grassmannian stochastic analysis and the stochastic quantization of Euclidean Fermions