21 research outputs found
Spatial birth-and-death processes with a finite number of particles
Spatial birth-and-death processes with time dependent rates are obtained as
solutions to certain stochastic equations. The existence, uniqueness,
uniqueness in law and the strong Markov property of unique solutions are proven
when the integral of the birth rate over grows not
faster than linearly with the number of particles of the system. Martingale
properties of the constructed process provide a rigorous connection to the
heuristic generator. We also study pathwise behavior of an aggregation model.
The probability of extinction and the growth rate of the number of particles
conditioning on non-extinction are estimated.Comment: arXiv admin note: substantial text overlap with arXiv:1502.06783. New
version note: significant structural and other change
The maximal displacement of radially symmetric branching random walk in
We consider discrete-time branching random walks with a radially symmetric
distribution. Independently of each other individuals generate offspring whose
relative locations are given by a copy of a radially symmetric point process
. The number of particles at time form a supercritical
Galton-Watson process. We investigate the maximal distance to the origin of
such branching random walks. Conditioned on survival, we show that, under some
assumptions on , it grows in the same way as for branching
Brownian motion or a broad class of one-dimensional branching random walks: the
first term is linear in time and the second logarithmic. The constants in front
of these terms are explicit and depend only on the mean measure of
and dimension. Our main tool in the proof is a ballot theorem
with moving barrier which may be of independent interest
The continuous-time frog model can spread arbitrarily fast
The aim of the paper is to demonstrate that the continuous-time frog model can spreadarbitrary fast. The set of sites visited by an active particle can become infinite in a finitetime
Shape theorem for a one-dimensional growing particle system with a bounded number of occupants per site
We consider a one-dimensional discrete-space birth process with a bounded
number of particle per site. Under the assumptions of the finite range of
interaction, translation invariance, and non-degeneracy, we prove a shape
theorem. We also derive a limiting estimate and an exponential estimate on the
fluctuations of the position of the rightmost particle.Comment: 17 pages; a short section devoted to numerical simulations is added;
other small changes and improvement
Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate
We consider the speed of propagation of a {continuous-time continuous-space}
branching random walk with the additional restriction that the birth rate at
any spatial point cannot exceed . The dispersion kernel is taken to have
density that decays polynomially as , . We show
that if , then the system spreads at a linear speed, {while for
the spread is faster than linear}. We also consider
the mesoscopic equation corresponding to the microscopic stochastic system. We
show that in contrast to the microscopic process, the solution to the
mesoscopic equation spreads exponentially fast for every .Comment: v2 update: A new result is added covering the case for
the microscopic model. Further remarks and heuristic comments are added,
including connections to other models. Many minor changes are mad