According to Conley-Zehnder's theorem, any periodic Hamiltonian ODE in R^2n has at least 2n+1 geometrically distinct periodic orbits. For a stochastically stationary Hamiltonian ODE, the set of periodic orbits yields a translation invariant random process. In this talk, I will discuss an ergodic theorem for the density of periodic orbits, and formulate some open questions which are the stochastic variants of Conley-Zehnder's theorem.