Rate of Estimation for the Stationary Distribution of Stochastic Damping Hamiltonian Systems with Continuous Observations

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 $(Z_t)_{tin[0,T]}=(X_t,Y_t)_{t in [0,T]}$. 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.