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Here, we develop the "teaser" from the previous talk, showing that the fast-interaction model has an interface-valued solution where the left and right sides of the interface evolve as cadlag processes (with jumps inward) until they meet, at which point the common single-point interface evolves as a one-dimensional diffusion related to, but not generally equal to, the particle motion.
Building on the framework from the previous talk, we study exchangeability and Poisson properties of the slow and fast interaction particle systems. In particular, we show the sense in which the slow-interaction particle system represents a measure-valued process on the real line and give a martingale characterization of this process. As a "teaser" for the next talk, we also suggest a sense in which the J1 convergence of the slow interaction model to the fast interaction model is a generalization of a space-time scaling of interfaces studied by Tribe (1995).
Within an abstract framework for lookdown particle systems, we construct systems with "slow" local interactions, of the type studied in B., 2002 (but with a more general underlying motion process) and "fast" local interactions of the type studied in Donnelly et al., 2000 (but with additional path properties). This unified approach allows us to prove a convergence theorem: as the "slow" location interactions are sped up to an infinite rate, the type processes converge in the J1 topology, particle by particle.
Mueller & Tribe, 1995 study a stochastic PDE that arises as the limit of a long-range voter model. B., 2002 shows that this model also has a countable representation as an infinite-density, Brownian, interacting particle system. Tribe, 1995 studies the interface behaviour of the SPDE under a Brownian space-time scaling showing that a one-point interface persists and evolves as a Brownian motion. Because this space-time scaling is equivalent to scaling the interaction rate in the particle model, we can use the particle system machinery of Donnelly et al., 2000 and B., 2002 to study these scaled interfaces from another perspective. This machinery provides simple, intuitive proofs of many of the properties of the interface, and we generalize Tribe's results to particle systems with more general particle motions.
The Moran model is a simple finite-population genetic particle model with matched births and deaths. In the large-population limit, it converges to the Fleming-Viot measure-valued diffusion process, a close relative of super Brownian motion. We discuss an ordered particle construction that resembles the Moran model but assigns each individual a "rank": when two particles interact, it is always the lower-rank particle that gives birth to an offspring and the higher-rank particle that dies to make room for this new individual. Unlike the Moran model, this ordered model is trivially extended to an infinite population size, and the resulting model embeds Moran models for all finite population sizes and carries, as its limiting empirical measure, the Fleming-Viot process. The result is an easily visualized construction of the Fleming-Viot process that demonstrates its relationship to the Moran model.
We consider an infinite collection of one-dimensional, interacting Brownian particles each with an associated "type" and "level". Particles interact at a rate determined by the local time at zero of the distance between them, and---when an interaction occurs---the higher-level particle copies the type of the lower-level one. With appropriate initial conditions, the process at any fixed time is conditionally a Poisson point process, and the marginal location/type measure of the conditional mean satisfies a stochastic pde obtained by Mueller and Tribe (1995) as a long-range voter model limit. (Joint work with Tom Kurtz.)
We begin by considering a genetic stepping stone model with sites on the one-dimensional lattice. Within each site, particles are subject to Moran model interactions with selection. Between interactions, they mutate and migrate independently. If the population density is held constant while the lattice density increases, then---under suitable parameter scalings---the migration random walks converge to Brownian motions and the limiting interactions are determined by Poisson counting processes driven by clocks proportional to the local times at zero of the distances between pairs. The result is a finite-density collection of Brownian motions with local-time Moran interactions.
We study the limiting behavior of these models as the population density increases to infinity by ordering the particles with randomly assigned "levels" in the non-negative reals. If neutral interactions are restricted to occur in only one direction, so that the higher-level particle changes its type to that of the lower-level particle, the result is an ordered model that can be extended to infinite densities.
Restricted to a given maximum level, these infinite-density ordered models have the same empirical location/type distributions as the original, symmetric Moran models. In the stepping-stone case, we establish this by means of a generator argument. In the Brownian case, we establish it through a more direct coupling.
Under appropriate initial conditions, these ordered models have a simple Poisson structure. In the Brownian case, for each fixed time, there exists a measure-valued diffusion nu such that the point process consisting of the location, type, and level of each particle is conditionally Poisson with mean measure nu(t) cross Lebesgue in location/type cross level space.
We study this diffusion process for the Brownian case with selection, showing that it almost surely has continuous paths and giving a martingale characterization.
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Last revised: Mon Feb 6 15:49:29 PST 2006