  Lyapunov exponents of dynamical systems are defined from the rates of
divergence of nearby trajectories. For stochastic systems, one typically
assumes that these trajectories are generated under the "same noise
realization". The purpose of this work is to critically examine what this
expression means. For Brownian particles, we consider two natural
interpretations of the noise: intrinsic to the particles or stemming from the
fluctuations of the environment. We show how they lead to different
distributions of the largest Lyapunov exponent as well as different fluctuating
hydrodynamics for the collective density field. We discuss, both at microscopic
and macroscopic levels, the limits in which these noise prescriptions become
equivalent. We close this paper by providing an estimate of the largest
Lyapunov exponent and of its fluctuations for interacting particles evolving
with the Dean-Kawasaki dynamics.
