Related papers: Stable fluctuations for ballistic random walks in …
This paper enhances the result of the work [G. Kozma, B. T\'oth, Ann. Probab. vol. 45 (2017) 4307-4347] . We prove the central limit theorem (in probability w.r.t. the environment) for the displacement of a random walker in divergence-free…
It is well known that the distribution of simple random walks on $\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on…
A zero-one law of Engelbert--Schmidt type is proven for the norm process of a transient random walk. An invariance principle for random walk local times and a limit version of Jeulin's lemma play key roles.
We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that…
We study the limit behaviour of a class of random walk models taking values in the $d$-dimensional unit standard simplex, $d\ge 1$, defined as follows. From an interior point $z$, the process chooses one of the $d+1$ vertices of the…
In this paper, we consider random walk in random environment on $\mathbb{Z}^{d}\,(d\geq1)$ and prove the Strassen's strong invariance principle for this model, via martingale argument and the theory of fractional coboundaries of Derriennic…
The position $x(t)$ of a particle diffusing in a one-dimensional uncorrelated and time dependent random medium is simply Gaussian distributed in the typical direction, i.e. along the ray $x=v_0 t$, where $v_0$ is the average drift. However,…
We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_\alpha>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then…
Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions $d\ge 2$. In 2002, Sznitman introduced for each $\gamma\in (0,1)$ the ballisticity conditions $(T)_\gamma$ and $(T'),$ the latter being defined as the…
We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the Central…
Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d larger than 1), are named Dirichlet when their step lengths are distributed according to a Dirichlet law. The latter continuous…
A reaction-diffusion model which is called the field-road model was introduced by Berestycki, Roquejoffre and Rossi [9] to describe biological invasion with fast diffusion on a line. In this paper, we investigate this model in a…
Modelling the propagation of a pulse in a dense {\em milieu} poses fundamental challenges at the theoretical and applied levels. To this aim, in this paper we generalize the telegraph equation to non-ideal conditions by extending the…
We consider random walks in a balanced random environment in $\mathbb{Z}^d$, $d\geq 2$. We first prove an invariance principle (for $d\ge2$) and the transience of the random walks when $d\ge 3$ (recurrence when $d=2$) in an ergodic…
We show that particle transport in a uniform, quantum multi-baker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semi-classical limit leads to diffusion. Random…
We consider a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal…
We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is…
We obtain Gaussian upper and lower bounds on the transition density q_t(x,y) of the continuous time simple random walk on a supercritical percolation cluster C_{\infty} in the Euclidean lattice. The bounds, analogous to Aronsen's bounds for…
In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition $(T')$, introduced by Sznitman [Ann. Probab. 29 (2001) 723--764] and by Schmitz [Ann. Inst. H. Poincar\'{e} Probab. Statist. 42…
We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required…