  We consider a system of non-interacting Brownian particles on a line with a
step-like initial condition, and we investigate the behavior of the local time
at the origin at large times. We compute the mean and the variance of the local
time, and we show that the memory effects are governed by the Fano factor
associated with the initial condition. For the uniform initial condition, we
show that the probability distribution of the local time admits a large
deviation form, and we compute the corresponding large deviation functions for
the annealed and quenched averaging schemes. The two resulting large deviation
functions are very different. Our analytical results are supported by extensive
numerical simulations.
