Branching Processes - A General Concept

Greven, Andreas; Rippl, Thomas; Gloede, Patrick

doi:10.30757/alea.v18-25

Cited by 5 publications

(20 citation statements)

References 35 publications

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“…We prove that the Í-valued Feller diffusion is in fact a branching process in a generalized sense; see [GRG18] for the concept and an alternative proof. It is shown that starting with two populations initially, the genealogies associated with these two populations evolve independently if we "restrict to the top of the tree".…”

Section: Basic Questionsmentioning

confidence: 97%

“…In the world of Fleming-Viot processes with fixed population sizes, this martingale problem approach has been used extensively on the space of ultrametric probability measure spaces, a closed subspace of Í, denoted by Í 1 ; see [GPW13,DGP12,GSW16]. Varying population sizes are considered in [Glö12,GRG18] and [GGR19].…”

Section: Basic Questionsmentioning

confidence: 99%

“…Some more technical points we collect in the appendix in Sections A, B, C and D. In Section A we give the calculation correcting the scaling limit result in [LN68]. In Sections B and C we collect some consequences of infinite divisibility on Í as studied in [GGR19] and [GRG18]. Finally, in Section D we briefly discuss approximation of solutions of certain martingale problems.…”

Section: Basic Questionsmentioning

confidence: 99%

“…This extends the concept of historical processes of Dawson and Perkins [DP91] suited for super processes on Ê d and allows to tackle general population processes; see [DG19] for a survey of our approach. As a result we can use recent work [GGR19,GRG18] to determine the Lévy measure, the excursion law, Evans branching with immigration from an immortal line and many other objects relevant for the genealogy-valued Feller-diffusion.…”

Section: Introductionmentioning

confidence: 99%

“…Lévy-Khintchine representation Finally we now relate the backbone representation with the CPP-representation via the Lévy-Khintchine representation of the (as we saw) infinitely divisible U * t , while U * ,+ t is only h-infinitely divisible if we merge the marks s ≤ t − h since the information contained in the color is also an information on the genealogy before time t − h (see [GRG18] for more on this). Different from (3.120), we look then for an i.i.d.…”

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Genealogy-valued Feller diffusion

Depperschmidt¹,

Greven²

2019

Preprint

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We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion model. In contrast to the approach via labeled trees in the continuum random tree world, the genealogies are modeled as equivalence classes of ultrametric measure spaces, elements of the space Í. This space is Polish and has a rich semigroup structure for the genealogy. We focus on the evolution of the genealogy in time and the large time asymptotics conditioned both on survival up to present time and on survival forever. We develop the calculus in such a way that it can be applied in the future to more complicated systems as logistic branching or state dependent branching.We prove existence, uniqueness and Feller property of solutions of the martingale problem for this genealogy valued, i.e. Í-valued Feller diffusion. We give the precise relation to the time-inhomogeneous Í 1 -valued Fleming-Viot process. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. Via the Feynman-Kac duality we deduce the generalized branching property of the Í-valued Feller diffusion. Using a semigroup operation on Í together with the branching property we obtain a Lévy-Khintchine formula for Í-valued Feller diffusion and determine explicitly the Lévy measure.For h > 0 we obtain a decomposition into depth-h subfamilies which leads to a representation of the in terms of a Cox point process of genealogies of single ancestor subfamilies. Furthermore, correcting a result from the Ê + -valued literature, we will identify the Í-valued process conditioned to survive until a finite time T . This is the key ingredient of the excursion law of the Í-valued Feller diffusion.We study long time asymptotics of the Í-valued Feller diffusion conditioned to survive forever, its generalized quasi-equilibrium and Kolmogorov-Yaglom limit law and show the limit processes solve well-posed Í-valued martingale problems. We also obtain various representations of the long time limits: backbone construction of the Palm distribution, the Í-valued version of the Kallenberg tree, the Í-valued version of Feller's branching diffusion with immigration from an immortal line à la Evans. This requires considering conditioned martingale problems which are of different form that those in the unconditioned case. On the level of Ívalued processes we still have equality (in law) of the Q-process, i.e., the process conditioned to survive up to time T in the limit T → ∞, the size-biased process and Evans' branching process with immigration from an immortal line. The Í-valued generalized quasi-equilibrium is a size-biased version of the Kolmogorov-Yaglom limit law.The above results are also key tools for analyzing genealogies in spatial branching populations. We construct the genealogy of the spatial version of the Feller diffusion on a countable group (super random walk). We give results on a martingale problem characterization, duality, generalized branching property and the long time behavior for this object. As an appl...

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Section: Basic Questionsmentioning

confidence: 97%

Section: Basic Questionsmentioning

confidence: 99%

Section: Basic Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Genealogy-valued Feller diffusion

Depperschmidt¹,

Greven²

2019

Preprint

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Stochastic Evolution of Genealogies of Spatial Populations: State Description, Characterization of Dynamics and Properties

Depperschmidt¹,

Greven²

2020

Genealogies of Interacting Particle Systems

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We survey results on the description of stochastically evolving genealogies of populations and marked genealogies of multitype populations or spatial populations via tree-valued Markov processes on (marked) ultrametric measure spaces. In particular we explain the choice of state spaces and their topologies, describe the dynamics of genealogical Fleming-Viot and branching models by well-posed martingale problems, and formulate the typical results on the longtime behavior. Furthermore we discuss the basic techniques of proofs and sketch as two key tools of analysis the different forms of duality and the Girsanov transformation.

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Branching trees I: concatenation and infinite divisibility

Glöde¹,

Greven²,

Rippl³

2019

Electron. J. Probab.

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The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by (equivalence classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space U which we recall. In order to then analyze the family structure in this coding we introduce an algebraic structure on um-spaces (a consistent collection of semigroups). This allows us to obtain a path of decompositions of subfamilies of fixed kinship h (described as ultrametric measure spaces), for every depth h as a measurable functional of the genealogy.Technically the elements of the semigroup are those um-spaces which have diameter less or equal to 2h called h-forests (h > 0). They arise from a given ultrametric measure space by applying maps called h−truncation. We can define a concatenation of two h-forests as binary operation. The corresponding semigroup is a Delphic semigroup and any h-forest has a unique prime factorization in h-trees (um-spaces of diameter less than 2h). Therefore we have a nested R + -indexed consistent (they arise successively by truncation) collection of Delphic semigroups with unique prime factorization.Random elements in the semigroup are studied, in particular infinitely divisible random variables. Here we define infinite divisibility of random genealogies as the property that the h-tops can be represented as concatenation of independent identically distributed h-forests for every h and obtain a Lévy-Khintchine representation of this object and a corresponding representation via a concatenation of points of a Poisson point process of h-forests.Finally the case of discrete and marked um-spaces is treated allowing to apply the results to both the individual based and most important spatial populations.The results have various applications. In particular the case of the genealogical (U-valued) Feller diffusion and genealogical (U V -valued) super random walk is treated based on the present work in [DG19b] and [GRG].In the part II of this paper we go in a different direction and refine the study in the case of continuum branching populations, give a refined analysis of the Laplace functional and give a representation in terms of a Cox process on h-trees, rather than forests.

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Branching Processes - A General Concept

Cited by 5 publications

References 35 publications

Genealogy-valued Feller diffusion

Genealogy-valued Feller diffusion

Stochastic Evolution of Genealogies of Spatial Populations: State Description, Characterization of Dynamics and Properties

Branching trees I: concatenation and infinite divisibility

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