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

Density estimates and short-time asymptotics for a hypoelliptic diffusion process