  We study experimentally the impact of inertial reference frame changes on
overdamped Brownian motion. The reference frame changes are implemented by
inducing, with a laser, laminar convection flows in a column of fluid where
Brownian microspheres are dispersed. The convection flow plays the role of the
relative velocity between the laboratory and the fluid comoving frames, and
enables us to analyse the consequences of Galilean transformations on Brownian
diffusion. We verify in particular how the Brownian dynamics remains "weakly"
Galilean invariant, in agreement with recent discussions. We also carefully
look at the consequences of Galilean relativity on the Brownian energetics.
This leads us to derive a Galilean invariant expression of the stochastic
thermodynamic first law, consistent with existing theoretical results. We
finally discuss a potential ambiguity of the Galilean relativistic features of
diffusive systems that has obvious practical implications in the context of
force measurements in external flows.
