Consider a large system of $N$ Brownian motions in $\R ^d$ fixed on a time interval $[0,β]$ with symmetrized initial and terminal conditions, under the influence of a trap potential. Such systems describe systems of bosons at positive temperatures confined in a spatial domain. We describe the large $N$ behavior of the averaged path (that is, their empirical path measure) and its connection with a well known optimal transport problem formulated by Erwin Schrödinger. We also explore the asymptotic behavior of the Brownian motions in terms of Large Deviations. In particular, the rate function that governs the mean of occupation measures turns out to be the well-known Donsker-Varadhan rate function. We therefore prove a simple formula for the large $N$ asymptotic of the symmetrized trace of $e^{-β\Hcal_N}$, where $\Hcal_N$ is an $N$ particle Hamilton operator in a trap