Recurrence for persistent random walks in two dimensions

@article{Lenci2007RecurrenceFP,
  title={Recurrence for persistent random walks in two dimensions},
  author={Marco Lenci},
  journal={Stochastics and Dynamics},
  year={2007},
  volume={07},
  pages={53-74}
}
  • M. Lenci
  • Published 20 July 2005
  • Mathematics
  • Stochastics and Dynamics
We discuss the question of recurrence for persistent, or Newtonian, random walks in ℤ2, i.e. random walks whose transition probabilities depend both on the walker's position and incoming direction. We use results by Toth and Schmidt–Conze to prove recurrence for a large class of such processes, including all "invertible" walks in elliptic random environments. Furthermore, rewriting our Newtonian walks as ordinary random walks in a suitable graph, we gain a better idea of the geometric features… 

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References

SHOWING 1-10 OF 18 REFERENCES

Symmetric random walks in random environments

We consider a random walk on thed-dimensional lattice ℤd where the transition probabilitiesp(x,y) are symmetric,p(x,y)=p(y,x), different from zero only ify−x belongs to a finite symmetric set

Persistent Random Walks in a One-Dimensional Random Environment

Recently it has become an intriguing problem to understand when random walks (r.w.) in a random environment (r.e.) converge to the Wiener process in the diffusion limit. Sinai's astonishing

A note on the central limit theorem for two-fold stochastic random walks in a random environment

We consider a class of two-fold stochastic random walks in a random environment. The transition probability is given by an ergodic random fi eld on Zd with two-fold stochastic realizations. The

On joint recurrence

Weak convergence of a random walk in a random environment

Let πi(x),i=1,...,d,x∈Zd, satisfy πi(x)≧α>0, and π1(x)+...+πd(x)=1. Define a Markov chain onZd by specifying that a particle atx takes a jump of +1 in theith direction with probability 1/2πi(x) and a

Typicality of recurrence for Lorentz gases

  • M. Lenci
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2006
It is a safe conjecture that most (not necessarily periodic) two-dimensional Lorentz gases with finite horizon are recurrent. Here we formalize this conjecture by means of a stochastic ensemble of

Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results

Random Walks in Random Environment

  • O. Zeitouni
  • History
    Encyclopedia of Complexity and Systems Science
  • 2009
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal,

Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications

  • J. Conze
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1999
Let $(E,{\cal A},\mu,T)$ be a dynamical system and let $\Phi$ be a function defined on $E$ with values in $\mathbb{R}^2$. We give a criterion, the central limit theorem along subsequences of positive

Persistent random walks in random environment

SummaryWeak convergence of a class of functionals of PRWRE is proved. As a consequence CLT is obtained for the normed trajectory.