No abstract

Let {S k , k ≥ 0} be a symmetric random walk on Z d , and {η(x), x ∈ Z d } an independent random field of centered i.i.d. random variables with tail decayWe consider a random walk in random scenery, that is X n = η(S 0 ) + · · · + η(S n ). We present asymptotics for the probability, over both randomness, that {X n > n β } for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process x l 2 n (x), where l n (x) is the number of visits of site x up to time n.

To cite this version:Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pene, Bruno Schapira. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Annals of Probability, Institute of Mathematical Statistics, 2011, pp.Vol. 39, No 6, 2079-2118.1214/10-AOP606>. A LOCAL LIMIT THEOREM FOR RANDOM WALKS IN RANDOM SCENERY AND ON RANDOMLY ORIENTED LATTICESFABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, FRANÇ OISE PÈNE, AND BRUNO SCHAPIRA Abstract. Random walks in random scenery are processes defined by Zn := n k=1 ξX 1 +...+X k , where (X k , k ≥ 1) and (ξy, y ∈ Z) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2] respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α = 1 and as n → ∞, of n −δ Zn, for some suitable δ > 0 depending on α and β. Here we are interested in the convergence, as n → ∞, of n δ P(Zn = ⌊n δ x⌋), when x ∈ R is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

We prove large deviations principles in large time, for the Brownian occupation time in random scenery 1 t t 0 ξ(B s ) ds. The random field is constant on the elements of a partition of R d into unit cubes. These random constants, say ξ(j ), j ∈ Z d consist of i.i.d. bounded variables, independent of the Brownian motion {B s , s ≥ 0}. This model is a timecontinuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in "quenched" and "annealed" settings.