  We consider incompressible generalized Newtonian fluids in two space
dimensions perturbed by an additive Gaussian noise. The velocity field of such
a fluid obeys a stochastic partial differential equation with fully nonlinear
drift due to the dependence of viscosity on the shear rate. In particular, we
assume that the extra stress tensor is of power law type, i.\,e. a polynomial
of degree $p-1$, $p \in (1,2)$, i.\,e. the shear thinning case. We prove that
the associated Kolmogorov operator $K$ admits at least one infinitesimally
invariant measure $\mu$ satisfying certain exponential moment estimates.
Moreover, $K$ is $L^2$-unique w.\,r.\,t. $\mu$ provided $p \in (p^\ast,2)$,
where $p^\ast$ is the second root of $p^3 - 8p^2 + 14p -6 =0$, approximately
$p^\ast \approx 1.60407$.
