Franchissement et temps local pour l'oscillateur harmonique

We consider the second order Stochastic Differential Equation dPtβ = Vtβ dt with P0β = p0, dVtβ = βVtβdt − βω2Ptβ + βdWt with V0β = v0, where W stands for a standard Wiener process and where ω is a real constant. It is well-known that Pβ converges, as β goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of Pβ at level u during the time interval [0, t] · (NtPβ (u)) to the local time of P(LtP (u)).