On the recurrence of some random walks in random environment

18 pages, 2 figuresThis work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely often, for any k. We also consider direct products of one-dimensional recurrent RWRE with another RWRE or with a RW. We point out the that models involving one-dimensional recurrent RWRE are more recurrent than the corresponding models involving simple symmetric walk

Limit theorems for one and two-dimensional random walks in random scenery

International audienceRandom walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in{\mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb Z}^d$ and $\mathbb R$ respectively. We suppose that the distributions of $X_1$ and $\xi_0$ belong to the normal basin of attraction of stable distribution of index $\alpha\in(0,2]$ and $\beta\in(0,2]$. When $d=1$ and $\alpha\ne 1$, a functional limit theorem has been established in \cite{KestenSpitzer} and a local limit theorem in \cite{BFFN}. In this paper, we establish the convergence of the finite-dimensional distributions and a local limit theorem when $\alpha=d$ (i.e. $\alpha = d=1$ or $\alpha=d=2$) and $\beta \in (0,2]$. Let us mention that functional limit theorems have been established in \cite{bolthausen} and recently in \cite{DU} in the particular case where $\beta=2$ (respectively for $\alpha=d=2$ and $\alpha=d=1$)

On the one-sided exit problem for stable processes in random scenery

International audienceWe consider the one-sided exit problem for stable LÈvy process in random scenery, that is the asymptotic behaviour for $T$ large of the probability $$\mathbb{P}\Big[ \sup_{t\in[0,T]} \Delta_t \leq 1\Big]$$ where $$\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$$ Here $W=(W(x))_{x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and $(L_t(x))_{x\in\mathbb{R},t\geq 0}$ the local time of a stable Lévy process with index $\alpha\in (1,2]$, independent from the process $W$. Our result confirms some physicists prediction by Redner and Majumdar

A local limit theorem for random walks in random scenery and on randomly oriented lattices

International audienceRandom walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $\alpha\in (0,2]$ and $\beta\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to \infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on $\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to \infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when x\in \RR is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results

On the local time of random processes in random scenery

International audienceRandom walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where basically $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that $X_1$ is \ZZ-valued, centered and with finite moments of all orders. We also assume that $\xi_0$ is \ZZ-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that $(n^{-3/4}Z_{[nt]},t\ge 0)$ converges in distribution as $n\to \infty$ toward some self-similar process $(\Delta_t,t\ge 0)$ called Brownian motion in random scenery. In a previous paper, we established that ${\mathbb P}(Z_n=0)$ behaves asymptotically like a constant times $n^{-3/4}$, as $n\to \infty$. 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 $\Delta$ admits a bi-continuous version of its local time process which is locally Hölder continuous of order $1/4-\delta$ and $1/6-\delta$, respectively in the time and space variables, for any $\delta>0$. In particular, this gives a new proof of the fact, previously obtained by Khoshnevisan, that the level sets of $\Delta$ 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

Non-homogeneous hidden Markov-switching models for wind time series

International audienceIn this paper we propose various Markov-switching auotoregressive models for bivariate time series which describe wind conditions at a single location. The main originality of the proposed models is that the hidden Markov chain is not homogeneous, its evolution depending on the past wind conditions. It is shown that they permit to reproduce complex features of wind time series such as non-linear dynamics and the multimodal marginal distributions

Transient random walk in with stationary orientations

International audienceIn this paper, we extend a result of Campanino and Pétritis [  (2003) 391–412]. We study a random walk in with random orientations. We suppose that the orientation of the th floor is given by , where is a stationary sequence of random variables. Once the environment fixed, the random walk can go either up or down or can stay in the present floor (but moving with respect to its orientation). This model was introduced by Campanino and Pétritis in [  (2003) 391–412] when the is a sequence of independent identically distributed random variables. In [  (2007) 815–826], Guillotin-Plantard and Le Ny extend this result to a situation where the orientations of the floors are independent but chosen with stationary probabilities (not equal to 0 and to 1). In the present paper, we generalize the result of [  (2003) 391–412] to some cases when is stationary. Moreover we extend slightly a result of [ (2007) 815–826]

A Berry Esseen result for the billiard transformation

We consider billiard systems in the two dimensional torus with convex obstacles. In this paper, we prove a rate of convergence in $n^{-{1\over 2}}$in the central limit theorem in the case of the billiard transformation. For one-dimensional functions, we control the maximal decay between the distribution functions.For multi-dimensional functions, we control the Prokhorov metric

Planar Lorentz gas walk in a random scenery

We consider the planar Lorentz gas with finite horizon. The random scenery is give by a sequence of iid square integrable random variables. To each obstacle we associate one of this random variable . We suppose that each time the particle hits an obstacle, it wins the amount given by the random variable associated to the obstacle. We are interesting in the asymptotic behaviour of this amount

Marches aléatoires et modèle de Lorentz : une approche de la théorie du chaos

International audienceOn présente les marches aléatoires simples symétriques en dimensions 1 et 2, ainsi qu'un modèle proposé par Lorentz pour le mouvement des électrons dans les métaux (simplifié en billard de Sinai). Nous verrons l'analogie entre ces deux modèles (le premier étant purement aléatoire et le second purement déterministe mais avec un aléa dû à l'imprécision des mesures des conditions initiales). Cette analogie apparaît en regardant les deux questions suivantes : la récurrence et le comportement du nombre de sites visités avant l'instant n