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In this article we are concerned with finite dimensional Fermions, by whichwe mean vectors in a finite dimensional complex space embedded in the exterioralgebra over itself. These Fermions are spinless but possess the characterizinganticommutativity property. We associate invariant complex vector fields on theLie group $\mathrm{Spin}(2n+1)$ to the Fermionic creation and annihilationoperators. These vector fields are elements of the complexification of theregular representation of the Lie algebra $\mathfrak{so}(2n+1)$. As such, theydo not satisfy the canonical anticommutation relations, however, once they havebeen projected onto an appropriate subspace of $L^2(\mathrm{Spin}(2n+1))$,these relations are satisfied. We define a free time evolution of this systemof Fermions in terms of a symmetric positive-definite quadratic form in thecreation-annihilation operators. The realization of Fermionic creation andannihilation operators brought by the (invariant) vector fields allows us tointerpret this time evolution in terms of a positive selfadjoint operator whichis the sum of a second order operator, which generates a stochastic diffusionprocess, and a first order complex operator, which strongly commutes with thesecond order operator. A probabilistic interpretation is given in terms of aFeynman-Kac like formula with respect to the diffusion process associated withthe second order operator.

Finite dimensional systems of free Fermions and diffusion processes on Spin groups