Joint work with Clémentine PRIEUR (UJF) In this talk we consider the nonlinear harmonic oscillator given by the system of Stochastic Differential Equations d x_t = y_tdt d y_t = \sigma I dWt -(c(x_t ; y_t)y_t + \grad V (x_t))dt (0.1) We define (Z_t := (x_t ; y_t) _in R^(2d) ; t \ge 0) Where c and V are the damping force and the potential respectively. We assume that these functions verify the following hypothesis. Hypothesis H1 : (i) the potential V is lower bounded, smooth over R^d, V and \grad V have polynomial growth at infinity ; (ii) the damping coefficient c(x ; y) is smooth, has polynomial growth at infinity, and for all N > 0 : sup_x\le N ;y \in R^d \|c(x ; y)\|_H.S < \infty, and there exist c,L > 0 so that c^s(x, y)\ge cI > 0, \forall (|x| > L, y \in R^d). In two articles written by Wu (2001) and Talay (2001) it was shown that the Markov process Z_t has a unique invariant measure and that is exponentially ergodic, in some space of continuous functions. By using this result one can establish the weakly dependence of such a process. Then this property leads us naturally to built an asymptotically consistent kernel estimator of the density of the invariant measure, by using the observation of the process over a discrete grid. We can also prove that this estimator satisfies a CLT. Afterwards we consider an estimator of the variance of the noise \sigma^2, showing consistence and asymptotical normality. One of the difficulties in both procedures is that we only observe the coordinate x_t. Finally we study the asymptotic behavior of the number of crossing of the discrete process.