  We develop a theory of fluctuations for Brownian systems with weak long-range
interactions. For these systems, there exists a critical point separating a
homogeneous phase from an inhomogeneous phase. Starting from the stochastic
Smoluchowski equation governing the evolution of the fluctuating density field,
we determine the expression of the correlation function of the density
fluctuations around a spatially homogeneous equilibrium distribution. In the
stable regime, we find that the temporal correlation function of the Fourier
components of the density fluctuations decays exponentially rapidly with the
same rate as the one characterizing the damping of a perturbation governed by
the mean field Smoluchowski equation (without noise). On the other hand, the
amplitude of the spatial correlation function in Fourier space diverges at the
critical point $T=T_{c}$ (or at the instability threshold $k=k_{m}$) implying
that the mean field approximation breaks down close to the critical point and
that the phase transition from the homogeneous phase to the inhomogeneous phase
occurs sooner. By contrast, the correlations of the velocity fluctuations
remain finite at the critical point (or at the instability threshold). We give
explicit examples for the Brownian Mean Field (BMF) model and for Brownian
particles interacting via the gravitational potential and via the attractive
Yukawa potential. We also introduce a stochastic model of chemotaxis for
bacterial populations generalizing the mean field Keller-Segel model by taking
into account fluctuations.
