  Galilean invariance is a cornerstone of classical mechanics. It states that
for closed systems the equations of motion of the microscopic degrees of
freedom do not change under Galilean transformations to different inertial
frames. However, the description of real world systems usually requires
coarse-grained models integrating complex microscopic interactions
indistinguishably as friction and stochastic forces, which intrinsically
violate Galilean invariance. By studying the coarse-graining procedure in
different frames, we show that alternative rules -- denoted as "weak Galilean
invariance" -- need to be satisfied by these stochastic models. Our results
highlight that diffusive models in general can not be chosen arbitrarily based
on the agreement with data alone but have to satisfy weak Galilean invariance
for physical consistency.
