Density Estimates for a Random Noise Propagating through a Chain of Di erential Equations

We here provide two sided bounds for the density of the solution of a system of n di erential equations of dimension d, the rst one being forced by a non-degenerate random noise and the n− 1 other ones being degenerate. The system formed by the n equations satis es a suitable Hörmander condition: the second equation feels the noise plugged into the rst equation, the third equation feels the noise transmitted from the rst to the second equation and so on..., so that the noise propagates one way through the system. When the coe cients of the system are Lipschitz continuous, we show that the density of the solution satis es Gaussian bounds with non-di usive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem.