33 research outputs found
Critical branching Brownian motion with absorption: survival probability
We consider branching Brownian motion on the real line with absorption at
zero, in which particles move according to independent Brownian motions with
the critical drift of . Kesten (1978) showed that almost surely this
process eventually dies out. Here we obtain upper and lower bounds on the
probability that the process survives until some large time . These bounds
improve upon results of Kesten (1978), and partially confirm nonrigorous
predictions of Derrida and Simon (2007)
Critical branching Brownian motion with absorption: particle configurations
We consider critical branching Brownian motion with absorption, in which
there is initially a single particle at , particles move according to
independent one-dimensional Brownian motions with the critical drift of
, and particles are absorbed when they reach zero. Here we obtain
asymptotic results concerning the behavior of the process before the extinction
time, as the position of the initial particle tends to infinity. We
estimate the number of particles in the system at a given time and the position
of the right-most particle. We also obtain asymptotic results for the
configuration of particles at a typical time
An integral test for the transience of a Brownian path with limited local time
We study a one-dimensional Brownian motion conditioned on a self-repelling
behaviour. Given a nondecreasing positive function f(t), consider the measures
mu_t obtained by conditioning a Brownian path so that L_s< f(s), for all s<t,
where L_s is the local time spent at the origin by time s. It is shown that the
measures mu_t are tight, and that any weak limit of mu_t as t tends to infinity
is transient provided that t^{-3/2}f(t) is integrable. We conjecture that this
condition is sharp and present a number of open problems.Comment: 3 figures. Some typos corrected
Gibbs distributions for random partitions generated by a fragmentation process
In this paper we study random partitions of 1,...n, where every cluster of
size j can be in any of w\_j possible internal states. The Gibbs (n,k,w)
distribution is obtained by sampling uniformly among such partitions with k
clusters. We provide conditions on the weight sequence w allowing construction
of a partition valued random process where at step k the state has the Gibbs
(n,k,w) distribution, so the partition is subject to irreversible fragmentation
as time evolves. For a particular one-parameter family of weight sequences
w\_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent
process with affine collision rate K\_{i,j}=a+b(i+j) for some real numbers a
and b. Under further restrictions on a and b, the fragmentation process can be
realized by conditioning a Galton-Watson tree with suitable offspring
distribution to have n nodes, and cutting the edges of this tree by random
sampling of edges without replacement, to partition the tree into a collection
of subtrees. Suitable offspring distributions include the binomial, negative
binomial and Poisson distributions.Comment: 38 pages, 2 figures, version considerably modified. To appear in the
Journal of Statistical Physic
The random walk penalised by its range in dimensions
We study a self-attractive random walk such that each trajectory of length
is penalised by a factor proportional to , where is
the set of sites visited by the walk. We show that the range of such a walk is
close to a solid Euclidean ball of radius approximately ,
for some explicit constant . This proves a conjecture of Bolthausen
who obtained this result in the case .Comment: Revised version, local errors and typos correcte