In 1962, Dyson showed that the spectrum of a $n\times n$ random Hermitian matrix, whose entries (real and imaginary) diffuse according to $n^2$ independent Ornstein-Uhlenbeck processes, evolves as $n$ non-colliding Brownian particles held together by a drift term. When $n\to \infty$, the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend...

Dans cet article, nous pénalisons la loi d’une araignée brownienne ${\left(A_t\right)}_{t\ge 0}$ prenant ses valeurs dans un ensemble fini $E$ de demi-droites concourantes, avec un poids égal à $\frac{1}{Z_t}exp({\alpha }_{N_t}{X}_t+\gamma L_t)$, où $t$ est un réel positif, ${\left({\alpha }_k\right)}_{k\in E}$ une famille de réels indexés par $E$, $\gamma$ un paramètre réel, $X_t$ la distance de $A_t$ à l’origine, $N_t$ ($\in E$) la demi-droite sur laquelle se trouve $A_t$, $L_t$ le temps local de ${\left(X_s\right)}_{0\le s\le t}$ à l’origine, et $Z_t$ la constante de normalisation. Nous montrons que la famille des mesures de probabilité obtenue par ces pénalisations converge vers...

Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in...

Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in...

Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p>1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds−μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p. Here f bounded or bounded and compactly supported and A(f)=‖f‖∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under...

Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc et al. [Stoc. Proc. Appl. 119, (2009) 897–923] introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using the regeneration method of Nummelin, extended to the time-continuous...

La marche aléatoire (ou marche au hasard) est un objet fondamental de la théorie des probabilités. Un des problèmes les plus intéressants pour la marche aléatoire (ainsi que pour le mouvement brownien, son analogue dans un contexte continu) est de savoir comment elle recouvre des ensembles où se trouvent les points qui sont souvent (ou au contraire, rarement) visités, et combien il y a de tels points. Les travaux de Dembo, Peres, Rosen et Zeitouni permettent de résoudre plusieurs conjectures importantes...

We consider one-dimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.58... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). We also study the random walk case and show that the process...

Let ξ(k, n) be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process ξ(k, n)−ξ(0, n) in terms of a brownian sheet and an independent Wiener process (brownian motion), time changed by an independent brownian local time. Some related results and consequences are also established.

We consider a semidynamical system $(X,ℬ,\Phi ,w)$. We introduce the cone $𝔸$ of continuous additive functionals defined on $X$ and the cone $𝒫$ of regular potentials. We define an order relation “$\le$” on $𝔸$ and a specific order “$\prec$” on $𝒫$. We will investigate the properties of $𝔸$ and $𝒫$ and we will establish the relationship between the two cones.

Let $(B_t,t\in [0,1])$ be a linear Brownian motion starting from 0 and denote by $(L_t\left(x\right),t\ge 0,x\in ℝ)$ its local time. We prove that the spatial trajectories of the Brownian local time have the same Besov-Orlicz regularity as the Brownian motion itself (i.e. for all t>0, a.s. the function $x\to L_t\left(x\right)$ belongs to the Besov-Orlicz space $B_{M_2,\infty }^{1/2}$ with $M_2\left(x\right)=e^{{\left|x\right|}^2}-1$). Our result is optimal.