  We consider a gas of independent Brownian particles on a bounded interval in
contact with two particle reservoirs at the endpoints. Due to the Brownian
nature of the particles, infinitely many particles enter and leave the system
in each time interval. Nonetheless, the dynamics can be constructed as a Markov
process with continuous paths on a suitable space. If $\lambda_0$ and
$\lambda_1$ are the chemical potentials of the boundary reservoirs, the
stationary distribution (reversible if and only if $\lambda_0=\lambda_1$) is a
Poisson point process with intensity given by the linear interpolation between
$\lambda_0$ and $\lambda_1$. We then analyze the empirical flow that it is
defined by counting, in a time interval $[0,t]$, the net number of particles
crossing a given point $x$. In the stationary regime we identify its statistics
and show that it is given, apart an $x$ dependent correction that is bounded
for large $t$, by the difference of two independent Poisson processes with
parameters $\lambda_0$ and $\lambda_1$.
