6,495 research outputs found

### Weak convergence for the minimal position in a branching random walk: a simple proof

Consider the boundary case in a one-dimensional super-critical branching
random walk. It is known that upon the survival of the system, the minimal
position after $n$ steps behaves in probability like ${3\over 2} \log n$ when
$n\to \infty$. We give a simple and self-contained proof of this result, based
exclusively on elementary properties of sums of i.i.d. real-valued random
variables.Comment: corrected reference in introductio

### Favourite sites of simple random walk

We survey the current status of the list of questions related to the
favourite (or: most visited) sites of simple random walk on Z, raised by Pal
Erdos and Pal Revesz in the early eighties.Comment: survey paper, 14 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

### A subdiffusive behaviour of recurrent random walk in random environment on a regular tree

We are interested in the random walk in random environment on an infinite
tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion.
Our paper focuses on the almost sure asymptotic behaviours of a recurrent
random walk $(X\_n)$ in random environment on a regular tree, which is closely
related to Mandelbrot [13]'s multiplicative cascade. We prove, under some
general assumptions upon the distribution of the environment, the existence of
a new exponent $\nu\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$
behaves asymptotically like $n^{\nu}$. The value of $\nu$ is explicitly
formulated in terms of the distribution of the environment.Comment: 29 pages with 1 figure. Its preliminary version was put in the
following web site:
http://www.math.univ-paris13.fr/prepub/pp2005/pp2005-28.htm

### 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

### Aggregation rates in one-dimensional stochastic systems with adhesion and gravitation

We consider one-dimensional systems of self-gravitating sticky particles with
random initial data and describe the process of aggregation in terms of the
largest cluster size L_n at any fixed time prior to the critical time. The
asymptotic behavior of L_n is also analyzed for sequences of times tending to
the critical time. A phenomenon of phase transition shows up, namely, for small
initial particle speeds (``cold'' gas) L_n has logarithmic order of growth
while higher speeds (``warm'' gas) yield polynomial rates for L_n.Comment: Published at http://dx.doi.org/10.1214/009117904000000900 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### A weakness in strong localization for Sinai's walk

Sinai's walk is a recurrent one-dimensional nearest-neighbor random walk in
random environment. It is known for a phenomenon of strong localization,
namely, the walk spends almost all time at or near the bottom of deep valleys
of the potential. Our main result shows a weakness of this localization
phenomenon: with probability one, the zones where the walk stays for the most
time can be far away from the sites where the walk spends the most time. In
particular, this gives a negative answer to a problem of Erd\H{o}s and
R\'{e}v\'{e}sz [Mathematical Structures--Computational
Mathematics--Mathematical Modelling 2 (1984) 152--157], originally formulated
for the usual homogeneous random walk.Comment: Published at http://dx.doi.org/10.1214/009117906000000863 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Slower deviations of the branching Brownian motion and of branching random walks

We have shown recently how to calculate the large deviation function of the
position $X_{\max}(t)$ of the right most particle of a branching Brownian
motion at time $t$. This large deviation function exhibits a phase transition
at a certain negative velocity. Here we extend this result to more general
branching random walks and show that the probability distribution of
$X_{\max}(t)$ has, asymptotically in time, a prefactor characterized by non
trivial power law.Comment: 13 pages and 1 figur

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