On the local time of random processes in random scenery

TL;DR

This paper studies the local time behavior of random walks in random scenery, providing precise asymptotics for return probabilities, establishing local time regularity, and confirming Hausdorff dimension results for the process's level sets.

Contribution

It extends the local limit theorem for random walks in random scenery to multiple times and analyzes the local time's regularity and Hausdorff dimension of level sets.

Findings

Precise asymptotics for the probability of simultaneous returns to zero.

Existence of a bi-continuous local time with specific Hölder continuity.

Convergence of moments of normalized local time to the continuous limit.

Abstract

Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where basically $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ 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 $\xi_0$ 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 $(\Delta_t,t\ge 0)$ 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 $\Delta$ admits a bi-continuous version of its local time process which is locally H\"older 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.