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...
Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope − , where denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when → 0, this probability decays like exp{−(+o(1)) / 1/2}, where is a positive constant depending...
We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see
(1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE...
These notes provide an elementary and self-contained introduction to branching random
walks.
Section 1 gives a brief overview of Galton–Watson trees, whereas Section 2 presents the
classical law of large numbers for branching random walks. These two short sections are
not exactly indispensable, but they introduce the idea of using size-biased trees, thus
giving motivations and an avant-goût to the main part, Section 3, where branching random
...
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