  We establish an exact mapping between (i) the equilibrium (imaginary time)
dynamics of non-interacting fermions trapped in a harmonic potential at
temperature $T=1/\beta$ and (ii) non-intersecting Ornstein-Uhlenbeck (OU)
particles constrained to return to their initial positions after time $\beta$.
Exploiting the determinantal structure of the process we compute the universal
correlation functions both in the bulk and at the edge of the trapped Fermi
gas. The latter corresponds to the top path of the non-intersecting OU
particles, and leads us to introduce and study the time-periodic Airy$_2$
process, ${\cal A}^b_2(u)$, depending on a single parameter, the period $b$.
The standard Airy$_2$ process is recovered for $b=+\infty$. We discuss
applications of our results to the real time quantum dynamics of trapped
fermions.
