  We study the problem of the non-parametric estimation for the density $\pi$
of the stationary distribution of a stochastic two-dimensional damping
Hamiltonian system $(Z_t)_{t\in[0,T]}=(X_t,Y_t)_{t \in [0,T]}$. From the
continuous observation of the sampling path on $[0,T]$, we study the rate of
estimation for $\pi(x_0,y_0)$ as $T \to \infty$. We show that kernel based
estimators can achieve the rate $T^{-v}$ for some explicit exponent $v \in
(0,1/2)$. One finding is that the rate of estimation depends on the smoothness
of $\pi$ and is completely different with the rate appearing in the standard
i.i.d.\ setting or in the case of two-dimensional non degenerate diffusion
processes. Especially, this rate depends also on $y_0$. Moreover, we obtain a
minimax lower bound on the $L^2$-risk for pointwise estimation, with the same
rate $T^{-v}$, up to $\log(T)$ terms.
