The almost sure limits of the minimal position and the additive martingale in a branching random walk

Consider a real-valued branching random walk in the boundary case. Using the techniques developed by A\"id\'ekon and Shi [5], we give two integral tests which describe respectively the lower limits for the minimal position and the upper limits for the associated additive martingale.Comment: Revised version for Journal of Theoretical Probabilit

How big is the minimum of a branching random walk?

Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \to \infty$, $M_n- {3 \over 2} \log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the upper limits of $M_n$: upon the system's non-extinction, $\limsup\_{n\to \infty} {1\over \log \log \log n} ( M_n - {3\over2} \log n) = 1$ almost surely. We also study the problem of moderate deviations of $M_n$: $p(M_n- {3 \over 2} \log n > \lambda)$ for $\lambda\to \infty$ and $\lambda=o(\log n)$. This problem is closely related to the small deviations of a class of Mandelbrot's cascades

The slow regime of randomly biased walks on trees

We are interested in the randomly biased random walk on the supercritical Galton--Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_n)$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^3$ in the first $n$ steps. We study the localization problem of $X_n$ and prove that the quenched law of $X_n$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_n|}{(\log n)^2}$ converges in law to some non-degenerate limit on $(0, \infty)$ whose law is explicitly computed.Comment: 43 pages. We added a recent work by Jim Pitman ([38]) for the limiting la

The most visited sites of biased random walks on trees

We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical Galton--Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erd\H{o}s and R\'ev\'esz [11] for simple symmetric random walk on the line.Comment: 17 page

Moderate deviations for diffusions with Brownian potentials

We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani's lemma and Lamperti's representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609] who studied the corresponding problem for Sinai's random walk in random environment.Comment: Published at http://dx.doi.org/10.1214/009117904000000829 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Strong disorder implies strong localization for directed polymers in a random environment

In this note we show that in any dimension $d$, the strong disorder property implies the strong localization property. This is established for a continuous time model of directed polymers in a random environment : the parabolic Anderson Model.Comment: Accepted for publication in ALE