# Estimation for Stochastic Damping Hamiltonian

vendredi 29 avril 2011, 14h30 - 15h30

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.