Existence of the density for a singular jump process and its short time properties (Q2772936)

Consider the solution of a canonical stochastic differential equation NEWLINE\[NEWLINEdx_t(x)=\sum_{j=1}^m X_j(x_{t-}(x)) \circ dz_j(t),NEWLINE\]NEWLINE where \(z(t)\) is a Lévy process satisfying some non degeneracy and approximate scaling conditions; assume that the vector fields \(X_j\) satisfy the Hörmander condition. The author first states the existence of a density for \(x_t\); the proof relies heavily on a paper by H. Kunita and J. Oh which has not appeared yet. Then, taking this existence for granted, the author derives upper bounds for the density at two types of points, namely the accessible and asymptotically accessible points.