  On a $d$-dimensional Riemannian, spin manifold $(M,g)$ we consider
non-linear, stochastic partial differential equations for spinor fields, driven
by a Dirac operator and coupled to an additive Gaussian, vector-valued white
noise. We extend to the case in hand a procedure, introduced in [DDRZ20] for
the scalar counterpart, which allows to compute at a perturbative level the
expectation value of the solutions as well as the associated correlation
functions accounting intrinsically for the underlying renormalization freedoms.
This framework relies strongly on tools proper of microlocal analysis and it is
inspired by the algebraic approach to quantum field theory. As a concrete
example we apply it to a stochastic version of the Thirring model proving in
particular that it lies in the subcritical regime if $d\leq 2$.
