We study the fluctuations in equilibrium of a class of Brownian motions interacting through a potential. For a certain choice of exponential potential, the distribution of the system coincides with differences of free energies of the stationary semi-discrete or OConnell-Yor polymer. We show that for Gaussian potentials, the fluctuations are of order $N^{frac{1}{4}}$ when the time and system size coincide, whereas for a class of more general convex potentials $V$ the fluctuations are of order at most $N^{frac{1}{3}}$. In the OConnell-Yor case, we recover the known upper bounds for the fluctuation exponents using a dynamical approach, without reference to the polymer partition function interpretation.
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