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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
A new model for evolution in a spatial continuum
"... o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL ..."
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Cited by 20 (5 self)
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o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL
Continuumsites steppingstone models, coalescing exchangeable partitions, and random trees
, 1998
"... Analogues of steppingstone models are considered where the sitespace is continuous, the migration process is a general Markov process, and the type{space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectati ..."
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Cited by 16 (5 self)
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Analogues of steppingstone models are considered where the sitespace is continuous, the migration process is a general Markov process, and the type{space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuumsites steppingstone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to 1/2 and, moreover, this space is capacity equivalent to the middle1/2 Cantor set (and hence also to the Brownian zero set).
The spatial LambdaFlemingViot process: an eventbased construction and a lookdown representation
, 2013
"... We construct a measurevalued equivalent to the spatial ΛFlemingViot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To th ..."
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We construct a measurevalued equivalent to the spatial ΛFlemingViot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To this end, we use a particle representation which highlights the role of the genealogies in the attribution of types (or alleles) to the individuals of the population. This construction also enables us to clarify the statespace of the SLFV and to derive several path properties of the measurevalued process as well as of the labeled trees describing the genealogical relations between a sample of individuals. We complement it with a lookdown construction which provides a particle system whose empirical distribution at time t, seen as a process in t, has the law of the quenched SLFV. In all these results, the facts that we work with a fixed configuration of events and that reproduction occurs only locally in space introduce serious technical issues that are overcome by controlling the number of events occurring and of particles present in a given area over macroscopic time intervals.
COALESCING SYSTEMS OF NONBROWNIAN PARTICLES
"... Abstract. A wellknown result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a setvalued process by requiring particles to coalesce when they collide. Arratia noted tha ..."
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Abstract. A wellknown result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a setvalued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one. 1.
COALESCING SYSTEMS OF BROWNIAN PARTICLES ON THE SIERPINSKI GASKET AND STABLE PARTICLES ON THE LINE OR CIRCLE
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Steppingstone model with circular Brownian migration
, 2005
"... In this paper we consider a steppingstone model on a circle with circular Brownian migration. We first point out a connection between Arratia flow and the marginal distribution of this model. We then give a new representation for the steppingstone model using Arratia flow and circular coalescing B ..."
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In this paper we consider a steppingstone model on a circle with circular Brownian migration. We first point out a connection between Arratia flow and the marginal distribution of this model. We then give a new representation for the steppingstone model using Arratia flow and circular coalescing Brownian motion. Such a representation enables us to carry out some explicit computation. In particular, we find the Laplace transform for the time when there is only a single type left across the circle.