CROSSINGS AND LOCAL-TIMES FOR THE HARMONIC-OSCILLATOR

We consider the second order Stochastic Differential Equation dP(t)(be ta) = V-t(beta) dt With P-0(beta) = p(0), dV(t)(beta) = beta V-t(beta) dt - beta omega(2)P(t)(beta)dt + beta dW(t) with V-0(beta) = upsilon( 0), where W stands for a standard Wiener process and where omega is a real constant. It is well-known that P-beta converges, as beta goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of P-beta at level u during the time interval [0, t].(N-t(P beta)(u)) to the local time of P(L-t(P)(u)).