  We study a system of $n$ differential equations, each in dimension $d$. Only
the first equation is forced by a Brownian motion and the dependence structure
is such that, under a local weak H\"ormander condition, the noise propagates to
the whole system. We prove upper bounds for the transition density (heat
kernel) and its derivatives of any order. Then we give precise short-time
asymptotics of the density at a suitable central limit time scale. Both these
results account for the different non-diffusive scales of propagation in the
various components. Finally, we provide a valuation formula for short-maturity
at-the-money Asian basket options under correlated local volatility dynamics.
