Abstract
We present random walks in random sceneries as well as three related models: U-statistics indexed by random walks, a model of stratified media with inhomogeneous layers (random one-way streets) and the one-dimensional Lévy-Lorentz gas (random roundabouts on a line). We present in particular results obtained in collaboration with Castell, Guillotin-Plantard, Br. Schapira, Franke, Wendler, Aurzada, Bianchi and Lenci.
Highlights
The aim of this paper is to show that random walks in random sceneries are processes of multiple interests
At the end of the 1970's, random walks in random sceneries (RWRS) have been introduced and studied as a probabilistic model by Borodin [8, 9] and Kesten and Spitzer [27], ten years later by Bolthausen [7] for a two-dimensional version, and more recently by many authors
This probabilistic model is a famous example of model with long time dependence, which is stationary but non Markov under the annealed law, and Markov but not stationary under the quenched law, and which converges after normalization to a self similar process with stationary increments, which is neither a Lévy process, nor a fractional Brownian motion
Summary
The aim of this paper is to show that random walks in random sceneries are processes of multiple interests. When the observables are in the normal domain of attraction of a β-stable distribution with β < 2 (which is the natural context to consider when the observable is not square integrable), a totally dierent behaviour can occur with a limit similar to those of RWRS but involving a Lévy sheet instead of a Lévy process. In this model, a particle goes straight on the real line at unit speed and can only change its direction (with probability 1 2) when it reaches some positions (called roundabouts in the present paper), these positions having been randomly xed at the beginning, with independent and identically distributed gaps between two consecutive roundabouts. A key point is that this process can be mathematically described with the use of a RWRS
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