  We study the problem of the non-parametric estimation for the density of the
stationary distribution of the multivariate stochastic differential equation
with jumps (Xt) , when the dimension d is bigger than 3. From the continuous
observation of the sampling path on [0, T ] we show that, under anisotropic
Holder smoothness constraints, kernel based estimators can achieve fast
convergence rates. In particular , they are as fast as the ones found by
Dalalyan and Reiss [9] for the estimation of the invariant density in the case
without jumps under isotropic Holder smoothness constraints. Moreover, they are
faster than the ones found by Strauch [29] for the invariant density estimation
of continuous stochastic differential equations, under anisotropic Holder
smoothness constraints. Furthermore, we obtain a minimax lower bound on the
L2-risk for pointwise estimation, with the same rate up to a log(T) term. It
implies that, on a class of diffusions whose invariant density belongs to the
anisotropic Holder class we are considering, it is impossible to find an
estimator with a rate of estimation faster than the one we propose.
