On the local time of random processes in random scenery

Abstract

Random walks in random scenery are processes defined by $Z_{n}:=\sum_{k=1}^{n}\xi_{X_{1}+\cdots+X_{k}}$, where basically $(X_{k},k\ge1)$ and $(\xi_{y},y\in\mathbb{Z})$ are two independent sequences of i.i.d. random variables. We assume here that $X_{1}$ is $\mathbb{Z}$-valued, centered and with finite moments of all orders. We also assume that $\xi_{0}$ is $\mathbb{Z}$-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that $(n^{-3/4}Z_{[nt]},t\ge0)$ converges in distribution as $n\to\infty$ toward some self-similar process $(\Delta_{t},t\ge0)$ 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 $\Delta$ admits a bi-continuous version of its local time process which is locally Hölder 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.