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1.
We consider random walks in random environments on Zd. Under a transitivity hypothesis that is much weaker than the customary ellipticity condition, and assuming an absolutely continuous invariant measure on the space of the environments, we prove the ergodicity of the annealed process w.r.t. the dynamics “from the point of view of the particle”. This implies in particular that the environment viewed from the particle is ergodic. As an example of application of this result, we give a general form of the quenched Invariance Principle for walks in doubly stochastic environments with zero local drift (martingale condition). 相似文献
2.
The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on Zd, d≥1. The red particles jump at rate 1 and are in a Poisson equilibrium with density μ. The green particle also jumps at rate 1, but uses different transition kernels p′ and p″ depending on whether it sees a red particle or not. It is shown that, in the limit as μ→∞, the speed of the green particle tends to the average jump under p′. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in Kesten and Sidoravicius (2005) to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space–time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain. 相似文献
3.
M. Gromov 《Geometric And Functional Analysis》2003,13(1):73-146
((no abstract)) .
Submitted: January 2002, Final version: December 2002. 相似文献
4.
《Stochastic Processes and their Applications》2020,130(12):7463-7482
We introduce the notion of Random Walk in Changing Environment (RWCE) — a random walk on a weighted graph in which the weights may change between steps. RWCE’s generalize many known RW (e.g. reinforced RW, true SAW). We explore possible properties of RWCE’s, and provide criteria for recurrence and transience when the underlying graph is or a tree. We construct a RWCE on where conductances can only change from 1 to 2 (once) but nevertheless the walk is transient, and conjecture that such behavior cannot happen when the changes do not depend on the location of the RWCE. 相似文献
5.
Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d ?? 2. In 2002, Sznitman introduced for each ${\gamma\in (0, 1)}$ the ballisticity conditions (T) ?? and (T??), the latter being defined as the fulfillment of (T) ?? for all ${\gamma\in (0, 1)}$ . He proved that (T??) implies ballisticity and that for each ${\gamma\in (0.5, 1)}$ , (T) ?? is equivalent to (T??). It is conjectured that this equivalence holds for all ${\gamma\in (0, 1)}$ . Here we prove that for ${\gamma\in (\gamma_d, 1)}$ , where ?? d is a dimension dependent constant taking values in the interval (0.366, 0.388), (T) ?? is equivalent to (T??). This is achieved by a detour along the effective criterion, the fulfillment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes. 相似文献
6.
We give a new proof of the central limit theorem for one dimensional symmetric random walk in random environment. The proof
is quite elementary and natural. We show the convergence of the generators and from this we conclude the convergence of the
process. We also investigate the hydrodynamic limit (HDL) of one dimensional symmetric simple exclusion in random environment
and prove stochastic convergence of the scaled density field. The macroscopic behaviour of this field is given by a linear
heat equation. The diffusion coefficient is the same as that of the corresponding random walk.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
7.
Simple random walk on the line in random environment 总被引:2,自引:0,他引:2
Summary We obtain strong limiting bounds for the maximal excursion and for the maximum reached by a random walk in a random environment. Our results derive from a simple proof of Pólya's theorem for the recurrence of the random walk on the line. As applications, we obtain bounds for the number of visits of the random walk at the origin. 相似文献
8.
一类随机环境下随机游动的常返性 总被引:1,自引:0,他引:1
张玥 《纯粹数学与应用数学》2004,20(1):53-56
就一类平稳环境θ下随机流动{Xn}n∈z 建立相应的Markov-双链{ηn}n∈z ={(xn,Tnθ)}n∈z ,并给出在该平稳环境θ下{xn}n∈z 为常返链的条件. 相似文献
9.
《Indagationes Mathematicae》2022,33(5):1049-1060
We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices and are connected with probability that asymptotically behaves like with , where denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits similar properties as classical discrete long-range percolation models studied by Berger (2002) with regard to recurrence and transience of the random walk. Moreover, we address a question which is related to a conjecture by Heydenreich, Hulshof and Jorritsma (2017) for this graph. 相似文献
10.
We derive a quenched moderate deviations principle for the one-dimensional nearest random walk in random environment, where the environment is assumed to be stationary and ergodic. The approach is based on hitting time decomposition. 相似文献
11.
12.
We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself. 相似文献
13.
14.
GAO ZhiQiang School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics Complex Systems Ministry of Education Beijing China Laboratoire de Mathatiques et Applications des Mathmatiques Universit de Bretagne-Sud BP Vannes France 《中国科学 数学(英文版)》2010,(2)
Suppose that the integers are assigned i.i.d. random variables {(β gx , . . . , β 1x , α x )} (each taking values in the unit interval and the sum of them being 1), which serve as an environment. This environment defines a random walk {X n } (called RWRE) which, when at x, moves one step of length 1 to the right with probability α x and one step of length k to the left with probability β kx for 1≤ k≤ g. For certain environment distributions, we determine the almost-sure asymptotic speed of the RWRE and show that the chance of the RWRE deviating below this speed has a polynomial rate of decay. This is the generalization of the results by Dembo, Peres and Zeitouni in 1996. In the proof we use a large deviation result for the product of random matrices and some tail estimates and moment estimates for the total population size in a multi-type branching process with random environment. 相似文献
15.
This paper discusses several aspects of shift-coupling for random walk in random environment. 相似文献
16.
We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in that is an extension of a result of Bolthausen et al. (2003) [4]. We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the first coordinate of the speed of the random walk under some strong assumptions on the distribution of the environment. 相似文献
17.
A. Astrauskas 《Lithuanian Mathematical Journal》1989,29(4):301-313
Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 4, pp. 627–644, October–December, 1989. 相似文献
18.
In this article, we mainly discuss the asymptotic behavior for multi-dimensional continuous-time random walk in random environment with holding times. By constructing a renewal structure and using the point “environment viewed from the particle”, under General Kalikow's Condition, we show the law of large numbers (LLN) and central limit theorem (CLT) for the escape speed of random walk. 相似文献
19.
Alain-Sol Sznitman 《Probability Theory and Related Fields》1999,115(3):287-323
We consider a d-dimensional random walk in random environment for which transition probabilities at each site are either neutral or present
an effective drift “pointing to the right”. We obtain large deviation estimates on the probability that the walk moves in
a too slow ballistic fashion, both under the annealed and quenched measures. These estimates underline the key role of large
neutral pockets of the medium in the occurrence of slowdowns of the walk.
Received: 12 March 1998 / Revised version: 19 February 1999 相似文献