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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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Cited by 48 (3 self)
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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.
Reflection and coalescence between independent onedimensional Brownian paths
 Ann. Inst. Henri Poincaré 36 (2000) 509–545. MR 2002a:60139
, 2000
"... Take two independent onedimensional brownian motions (B t ; t 2 [0; 1]) and (fi t ; t 2 [0; 1]) with B0 = 0 and fi 0 = 0 (fi can be seen as running backwards in time). Define (fl t ; t 2 [0; 1]) as being the function that is obtained by reflecting B on fi. Then fl is still a Brownian motion. Simila ..."
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Cited by 34 (1 self)
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Take two independent onedimensional brownian motions (B t ; t 2 [0; 1]) and (fi t ; t 2 [0; 1]) with B0 = 0 and fi 0 = 0 (fi can be seen as running backwards in time). Define (fl t ; t 2 [0; 1]) as being the function that is obtained by reflecting B on fi. Then fl is still a Brownian motion. Similar and more general results (with families of coalescing Brownian motions) are also derived. They enable to give a precise definition (in terms of reflection) of the joint realisation of finite families of coalescing/reflecting brownian motions. Key words: Brownian motion, coalescence, reflection. MSC Class.: 60J65 1 Introduction The main goal of this paper is to derive some facts concerning reflection and coalescence between independent onedimensional Brownian motions. Several papers in recent years studied and used families of coalescing onedimensional random walks or Brownian motions. These families and their main properties have been initially (to our knowledge) studied by Richard Ar...
Flows, coalescence and noise
, 2002
"... We are interested in stationary "fluid" random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coal ..."
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Cited by 21 (3 self)
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We are interested in stationary "fluid" random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels. In an intermediate phase, for which there exists a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a subnoise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.
Planar aggregation and the coalescing Brownian flow
, 2008
"... We study a scaling limit associated to a model of planar aggregation. The model is obtained by composing certain independent random conformal maps. The evolution of harmonic measure on the boundary of the cluster is shown to converge to the coalescing Brownian flow. 1 ..."
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Cited by 8 (4 self)
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We study a scaling limit associated to a model of planar aggregation. The model is obtained by composing certain independent random conformal maps. The evolution of harmonic measure on the boundary of the cluster is shown to converge to the coalescing Brownian flow. 1
Superprocesses with Coalescing Brownian Spatial Motion as LargeScale Limits
, 2006
"... Abstract. A superprocess with coalescing spatial motion is constructed in terms of onedimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the superprocess with dependent spatial motion studied in Da ..."
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Cited by 6 (4 self)
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Abstract. A superprocess with coalescing spatial motion is constructed in terms of onedimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the superprocess with dependent spatial motion studied in Dawson et al. (2001) and
A Class of Stochastic Partial Differential Equations for Interacting Superprocesses on a Bounded Domain
"... A class of interacting superprocesses on R, called superprocesses with dependent spatial motion (SDSMs), were introduced and studied in Wang [32] and Dawson et al. [9]. In the present paper, we extend this model to allow particles moving in a bounded domain in R d with killing boundary. We show that ..."
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Cited by 1 (0 self)
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A class of interacting superprocesses on R, called superprocesses with dependent spatial motion (SDSMs), were introduced and studied in Wang [32] and Dawson et al. [9]. In the present paper, we extend this model to allow particles moving in a bounded domain in R d with killing boundary. We show that under a proper rescaling, a class of discrete SPDEs for the empirical measurevalued processes generated by branching particle systems subject to the same white noise converge in L 2 (Ω, F, P) to the SPDE for a SDSM on a bounded domain and the corresponding martingale problem for the SDSMs on a bounded domain is wellposed. 1
Theory of Stochastic Differential Equations An Overview and Examples Shinzo Watanabe
"... We consider Itô’s stochastic differential equation (SDE). First, we would review a standard theory under standard assumptions. Then we would see how such a standard theory should be modified under different and more general assumptions and how and what new notions need be introduced to discuss such ..."
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We consider Itô’s stochastic differential equation (SDE). First, we would review a standard theory under standard assumptions. Then we would see how such a standard theory should be modified under different and more general assumptions and how and what new notions need be introduced to discuss such modifications.
NOISES IN STOCHASTIC PROCESSES
"... 1 Noise stability and sensitivity for stochastic processes represented by noises in discrete time ..."
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1 Noise stability and sensitivity for stochastic processes represented by noises in discrete time