  In this note, we give a new proof of Liggett's theorem on the invariant
measures of independent particle systems from [Lig78] in the particular case of
independent drifted Brownian motions. This particular case has received a lot
of attention recently due to its applications for the analysis of the local
extrema of discrete Gaussian free field. The novelty of our proof is that it
identifies directly the expected Poisson Point Process with exponential
intensity without relying on the Choquet-Deny convolution equation $\mu *
P=\mu$ ([ChoquetDeny60,Deny60]).
