Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEmpirical processes for recurrent and transient random walks in random scenery
128
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,tin[0,1]}$ with begin{equation*} W_n(s,t):=sum_{k=1}^{lfloor ntrfloor}big(1_{{xi_{S_k}leq s}}-sbig) end{equation*} where $(xi_x, xinmathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{ninmathbb N}$ is a random walk evolving in $mathbb{Z}^d$, independent of the $xi$s. In Wendler (2016), the case where $(S_n)_{ninmathbb N}$ is a recurrent random walk in $mathbb{Z}$ such that $(n^{-frac 1alpha}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $alpha$, with $alphain(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{ninmathbb N}$ is either: a) a transient random walk in $mathbb{Z}^d$, b) a recurrent random walk in $mathbb{Z}^d$ such that $(n^{-frac 1d}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $din{1,2}$.
rate research
Read More
We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^kappa)$ from the origin, $kappain(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{ u_0})$ from the origin, $ u_0in (0,kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{ u_1}$ from the origin, $ u_1in (kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^ u)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.
We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time aymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.
This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $dge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $dge 3$.
We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos and den Hollander, dos Santos, Sidoravicius. We show the CLT for such a random walk, valid for all supercritical infection rates for the contact process environment.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
