The (standard) Brownian web is a collection of coalescing one- dimensional
Brownian motions, starting from each point in space and time. It arises as the
diffusive scaling limit of a collection of coalescing random walks. We show
that it is possible to obtain a nontrivial limiting object if the random walks
in addition branch with a small probability. We call the limiting object the
Brownian net, and study some of its elementary properties.Comment: Published in at http://dx.doi.org/10.1214/07-AOP357 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models and results connected to the Brownian web and net, including alternative topologies, population genetic models, true self-repelling motion, planar aggregation, drainage networks, oriented percolation, black noise and critical percolation. Some open questions are discussed at the end.MSC 2000. Primary: 82C21 ; Secondary: 60K35, 60D05.
We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting Wright-Fisher diffusions, used to model a population with resampling, selection and mutations. We use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law.
This paper studies variations of the usual voter model that favor types that
are locally less common. Such models are dual to certain systems of branching
annihilating random walks that are parity preserving. For both the voter models
and their dual branching annihilating systems we determine all homogeneous
invariant laws, and we study convergence to these laws started from other
initial laws.Comment: Published in at http://dx.doi.org/10.1214/07-AAP444 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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