2012/02/15 by Castell, Fabienne, Nadine Guillotin‐Plantard, Guillotin--Plantard, Nadine +4
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.1202.3251
openalex publication_date 2012/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Random walks in random scenery are processes defined by Zn:=∑k=1nξX1+...+Xk, where basically (Xk,k≥ 1) and (ξy,y∈\mathbb Z) are two independent sequences of i.i.d. random variables. We assume here that X1 is \ZZ-valued, centered and with finite moments of all orders. We also assume that ξ0 is \ZZ-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that (n-3/4Z[nt],t≥ 0) converges in distribution as n→ ∞ toward some self-similar process (Δt,t≥ 0) called Brownian motion in random scenery. In a previous paper, we established that \mathbb P(Zn=0) behaves asymptotically like a constant times n-3/4, as n→ ∞. 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.