On the local time of random processes in random scenery
Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^nξ_{X_1+...+X_k}$, where basically $(X_k,k\ge 1)$ and $(ξ_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that $X_1$ is $\ZZ$-valued, centered and with finite moments of all orders. We also assume that $ξ_0$ is $\ZZ$-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that $(n^{-3/4}Z_{[nt]},t\ge 0)$ converges in distribution as $n\to \infty$ toward some self-similar process $(Δ_t,t\ge 0)$ called Brownian motion in random scenery. In a previous paper, we established that ${\mathbb P}(Z_n=0)$ behaves asymptotically like a constant times $n^{-3/4}$, as $n\to \infty$. 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 $Δ$ admits a bi-continuous version of its local time process which is locally Hölder continuous of order $1/4-δ$ and $1/6-δ$, respectively in the time and space variables, for any $δ>0$. In particular, this gives a new proof of the fact, previously obtained by Khoshnevisan, that the level sets of $Δ$ 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.