ESTIMATION OF THE DENSITY OF THE SOLUTION OF THE ROBUST ZAKAI EQUATION

We study the behaviour in small time of the density of the robust Zaka i equation under the weak Hormander's hypothesis. We avoid large devia tions phenomena by supposing that the drift at the departure is equal to zero. We get an asymptotic expansion over the diagonal of the densi ty where the observation is involved. In the case where all the terms of that asymptotic expansion are equal to zero, a pathology which is r elated to the Bismut's condition, we give an estimation of the decay b y using a probabilistic analogous of the Gevrey methods.