135 research outputs found

### 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

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