ON THE LOCAL TIME OF RANDOM PROCESSES IN RANDOM SCENERY

Random walks in random scenery are processes defined by Z(n) := Sigma(n)(k=1) xi(X1+ ...+Xk), where basically (X-k, k >= 1) and (xi(y), y is an element of Z) are two independent sequences of i.i.d. random variables. We assume here that X-1 is Z-valued, centered and with finite moments of all orders. We also assume that xi(0) is Z-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that (n(-3/4)Z([nt]), t >= 0) converges in distribution as n --> infinity toward some self-similar process (Delta(t), t >= 0) called Brownian motion in random scenery. In a previous paper, we established that P(Z(n) = 0) behaves asymptotically like a constant times n(-3/4), as n --> infinity. We extend here this local limit theorem: we give a precise asymptotic result for the probability for Z to return to zero simultaneously at several times. As a byproduct of our computations, we show that Delta admits a bi-continuous version of its local time process which is locally Holder continuous of order 1/4 - delta and 1/6 - delta, respectively, in the time and space variables, for any delta > 0. In particular, this gives a new proof of the fact, previously obtained by Khoshnevisan, that the level sets of Delta have Hausdorff dimension a.s. equal to 1/4. We also get the convergence of every moment of the normalized local time of Z toward its continuous counterpart.