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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.
Central limit theorems on nilpotent Lie groups
"... Lindeberg theorem is derived on stratied nilpotent Lie groups, that is a normal convergence theorem for a triangular system of probability measures in case of bounded (homogeneous) moments of second order. Using necessary and sucient conditions for convergence of convolution semigroups of probabili ..."
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Cited by 6 (4 self)
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Lindeberg theorem is derived on stratied nilpotent Lie groups, that is a normal convergence theorem for a triangular system of probability measures in case of bounded (homogeneous) moments of second order. Using necessary and sucient conditions for convergence of convolution semigroups of probability measures on Lie groups a LindebergFeller theorem is proved on the Heisenberg group.
Precise Estimates for the Subelliptic Heat Kernel on Htype Groups
, 2009
"... We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups G of Htype. Specifically, we show that there exist positive constants C1, C2 and a polynomial correction function Qt on G such that d2 − C1Qte 4t d2 − ≤ pt ≤ C2Qte 4t where pt is the heat kernel, and ..."
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We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups G of Htype. Specifically, we show that there exist positive constants C1, C2 and a polynomial correction function Qt on G such that d2 − C1Qte 4t d2 − ≤ pt ≤ C2Qte 4t where pt is the heat kernel, and d the CarnotCarathéodory distance on G. We also obtain similar bounds on the norm of its subelliptic gradient ∇pt. Along the way, we record explicit formulas for the distance function d and the subriemannian geodesics of Htype groups. On donne des estimations précises des bornes supérieures et inférieures du noyau de la chaleur souselliptique sur les groupes de Lie nilpotents G de type H. Plus précisément, on montre qu’il existe des constantes positives C1 et C2, et une fonction polynomiale corrective Qt sur G telles que d2 − C1Qte 4t d2 −
LévyKhinchin formula and existence of densities for convolution semigroups
"... on symmetric spaces ..."
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Poincareé invariant Markov processes and Gaussian random fields on relativistic phasespaces
, 1997
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Gaussian Bounds For Derivatives Of Central Gaussian Semigroups On Compact Groups
"... . For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumption, we prove that these derivatives satisfy certain Gaussian ..."
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. For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumption, we prove that these derivatives satisfy certain Gaussian bounds. 1.
CENTRAL LIMIT THEOREMS FOR THE BROWNIAN MOTION ON LARGE UNITARY GROUPS
, 904
"... Abstract. In this paper, we are concerned with the large N limit of linear combinations of entries of Brownian motions on the group of N × N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time are concerned, giving rise to vari ..."
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Cited by 4 (0 self)
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Abstract. In this paper, we are concerned with the large N limit of linear combinations of entries of Brownian motions on the group of N × N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time are concerned, giving rise to various limit processes, in relation to the geometric construction of the unitary Brownian motion. As an application, we recover certain results about linear combinations of the entries of Haar distributed random unitary matrices.
On the hypoellipticity of subLaplacians on infinite dimensional compact groups.
, 2001
"... this paper we need the following denition. Denition 1.2 Let ( t ) t>0 be a convolution semigroup of measures. 1. We say that ( t ) t>0 has property (AC) if t is asbolutely continuous with respect to Haar measure, for all t > 0. 2. We say that ( t ) t>0 has property (CK) if it satises ..."
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Cited by 4 (3 self)
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this paper we need the following denition. Denition 1.2 Let ( t ) t>0 be a convolution semigroup of measures. 1. We say that ( t ) t>0 has property (AC) if t is asbolutely continuous with respect to Haar measure, for all t > 0. 2. We say that ( t ) t>0 has property (CK) if it satises (AC) and has a continuous density x 7! t (x) such that lim t!0 t log t (e) = 0: 2 Theorem 1.3 Let G be a compact connected group. Let L be the innitesimal generator of a symmetric Gaussian semigroup ( t ) t>0 . Let S be any topological space of continuous functions whose topology is not weaker than the uniform topology. 1. If G is not a Lie group, L is never B 0 (G)Shypoelliptic. 2. The operator L is L 1 (G)C(G)hypoelliptic if and only if ( t ) t>0 satises (AC). 3. Fix 1 p < +1 and assume that L is biinvariant. If L is L p (G)Shypoelliptic then ( t ) t>0 must satisfy (CK). Hypoellipticity properties are closely related to whether or not any harmonic distribution must be continuous. Denition 1.4 Let P be a leftinvariant dierential operator on G of nite order. Let A be a xed space of distributions. We say that P is Aregular if, for any domain , each U 2 B 0 (G) such that 8 2 B 0( ; U 2 A and Z PUd = 0; can be represented in by a continuous function. We will prove a version of Theorem 1.3 where hypoellipticity properties are replaced by the corresponding regularity conditions in the sense of Denition 1.4. Note that this denition requires the distribution U to have two dierent properties in (a) 8 2 B 0( 7 U 2 A, which is a regularity property; (b) U is a solution of PU = 0 in Condition (a) plays a crucial role in the present context. In the companion paper [8], we prove some strong hypoellipticity results under the h...