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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 45 (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.
Moment bounds for the Smoluchowski equation and their consequences
 Publ. Res. Inst. Math. Sci
, 1978
"... We prove L ∞ ( R d × [0, ∞) ) bounds on moments Xa: = ∑ m∈N ma fm(x, t) of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than (n+m) ( d(n)+d(m) ) , and the diffusion rate d(·) is n ..."
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Cited by 14 (6 self)
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We prove L ∞ ( R d × [0, ∞) ) bounds on moments Xa: = ∑ m∈N ma fm(x, t) of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than (n+m) ( d(n)+d(m) ) , and the diffusion rate d(·) is nonincreasing and satisfies m−b1 ≤ d(m) ≤ m−b2 for some b1 and b2 satisfying 0 ≤ b2 < b1 < ∞, then any weak solution satisfies Xa ∈ L∞ ( Rd × [0, T] ) ∩ L1 ( Rd × [0, T] ) for every a ∈ N and T ∈ (0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. 1
Global divergence of spatial coalescents
 In preparation
, 2008
"... A class of processes called spatial Λcoalescents was recently introduced by Limic and Sturm (2006). In these models particles perform independent random walks on some underlying graph G. In addition, particles on the same site merge randomly according to some given coalescing mechanism. The goal of ..."
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Cited by 7 (3 self)
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A class of processes called spatial Λcoalescents was recently introduced by Limic and Sturm (2006). In these models particles perform independent random walks on some underlying graph G. In addition, particles on the same site merge randomly according to some given coalescing mechanism. The goal of the current work is to obtain several asymptotic results for these processes. If G = Z d, and the coalescing mechanism is Kingman’s coalescent, then starting with N particles at the origin, the number of particles is of order (log ∗ N) d at any fixed time (where log ∗ is the inverse tower function). At sufficiently large times this number is of order (log ∗ N) d−2. Betacoalescents behave similarly, with log log N in place of log ∗ N. Moreover, it is shown that on any graph and for general Λcoalescent, starting with infinitely many particles at a single site, the total number of particles will remain infinite at all times, almost surely.
Coagulation, Diffusion and the Continuous Smoluchowski Equation
, 2009
"... Abstract. The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the co ..."
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Abstract. The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the continuous form of the equations. In dimension d ≥ 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the equations in [4]. The principal innovation is a correlationtype bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blowup of solutions of the equations. AMS Subject Classifications. Primary 74A25, Secondary 60K35. Key words and phrases. Continuous Smoluchowski Equation, Coagulating Brownian Particles.
Moment Bounds for the Solutions of the Smoluchowski Equation with Coagulation and Fragmentation
"... We prove various Lp (Rd×[0, T]) bounds on moments Xa(x, t): = ∑ m∈N mafm(x, t), (respectively ∫ ∞ 0 mafm(x, t)dm,) where fm is a solution of the discrete (respectively continuous) Smoluchowski coagulationfragmentation equations with diffusion. In a previous paper [HR1] we proved similar results for ..."
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Cited by 6 (3 self)
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We prove various Lp (Rd×[0, T]) bounds on moments Xa(x, t): = ∑ m∈N mafm(x, t), (respectively ∫ ∞ 0 mafm(x, t)dm,) where fm is a solution of the discrete (respectively continuous) Smoluchowski coagulationfragmentation equations with diffusion. In a previous paper [HR1] we proved similar results for all weak solution to the discrete Smoluchowski’s equation provided that there is no fragmentation and certain moments are bounded in suitable Lqspaces initially. In this paper we prove the corresponding results in the case of the continuous Smoluchowski’s equation. When there is also fragmentation, we need to assume that the solution f is regular in the sense that f can be approximated by solutions to Smoluchowski equation for which the coagulation and fragmentation coefficients are 0 when the cluster sizes are large. We also need suitable assumptions on the coagulation rates to avoid gelation. On the fragmentation rate β, we assume that supn supm≤ℓ β(m, n)/n < ∞ for every positive ℓ, and that there exist constants a0 ≥ 0 and c0 such that β(n, m) ≤ c0(n + m) a0 1
Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion
, 2012
"... We prove various decay bounds on solutions (fn: n> 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ℓ fn in terms of a suitable average of the moments of the initial data for every positive ℓ. As a consequence, we ca ..."
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Cited by 2 (0 self)
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We prove various decay bounds on solutions (fn: n> 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ℓ fn in terms of a suitable average of the moments of the initial data for every positive ℓ. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of L p (R d × [0, T]) norms of the moments Xa(x, t): = ∑ m∈N mafm(x, t), ( ∫ ∞ 0 mafm(x, t)dm in the case of continuous Smoluchowski’s equation,) for every p ∈ [1, ∞]. In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is nonincreasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function φ(n) that is closely related to the total increase of the diffusion coefficient in the interval (0, n]. 1
Gelation for MarcusLushnikov Process
, 2010
"... MarcusLushnikov Process is a simple mean field model of coagulating particles that converges to the homogeneous Smoluchowski equation in the large mass limit. If the coagulation rates grow sufficiently fast as the size of particles get large, giant particles emerge in finite time. This is known as ..."
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MarcusLushnikov Process is a simple mean field model of coagulating particles that converges to the homogeneous Smoluchowski equation in the large mass limit. If the coagulation rates grow sufficiently fast as the size of particles get large, giant particles emerge in finite time. This is known as gelation and such particles are known as gels. Gelation comes in different flavors; simple, instantaneous and complete. In the case of an instantaneous gelation, a giant particle is formed in a very short time. If all particles coagulate to form a single particle in a time interval that stays bounded as total mass gets large, then we have a complete gelation. In this article, we describe conditions which guarantee any of the three possible gelations with explicit bounds on the size of gels and the time of their creations. 1
Equilibrium Fluctuations for a Model of CoagulatingFragmenting planar Brownian Particles
, 2009
"... One of the main purposes of statistical mechanics is to explain the macroscopic behavior of various phenomena in terms of the statistics of their microscopic structures. Macroscopically we often have a PDE involving a small number of parameters. The microscopic description however involves a large n ..."
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One of the main purposes of statistical mechanics is to explain the macroscopic behavior of various phenomena in terms of the statistics of their microscopic structures. Macroscopically we often have a PDE involving a small number of parameters. The microscopic description however involves a large number of components that are evolving by either deterministic