Rate of estimation for the stationary distribution of stochastic damping hamiltonian systems with continuous observations.

Summary We study the problem of the non-parametric estimation for the density π of the stationary distribution of a stochastic two-dimensional damping Hamiltonian system (Z_t) t∈[0,T ] = (X_t, Y_t) t∈[0,T ]. From the continuous observation of the sampling path on [0, T ], we study the rate of estimation for π(x_0 , y_0) as T → ∞. We show that kernel based estimators can achieve the rate T^{−v} for some explicit exponent v ∈ (0, 1/2). One finding is that the rate of estimation depends on the smoothness of π 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.