arXiv subjects

Andreas Greven

Publications and source records attributed to Andreas Greven.

28 records · Page 2Linked to original sources

arXiv2012

Tree-valued Fleming-Viot dynamics with mutation and selection

The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.

math.PR↗

arXiv2011

Tree-valued resampling dynamics: Martingale Problems and applications

The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the "individuals" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming-Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled "individuals". We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming-Viot dynamics.

math.PR↗

arXiv2011

Duality for spatially interacting Fleming-Viot processes with mutation and selection

Consider a system $X = ((x_ξ(t)), ξ\in Ω_N)_{t \geq 0}$ of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space $(\CP(\I))^{Ω_N}$, where $\I$ is the type space, ${Ω_N}$ the geographic space is assumed to be a countable group and $\CP$ denotes the probability measures. We establish various duality relations for this process. These dualities are function-valued processes which are driven by a coalescing-branching random walk, that is, an evolving particle system which in addition exhibits certain changes in the function-valued part at jump times driven by mutation. In the case of a finite type space $\I$ we construct a set-valued dual process, which is a Markov jump process, which is very suitable to prove ergodic theorems which we do here. The set-valued duality contains as special case a duality relation for any finite state Markov chain. In the finitely many types case there is also a further tableau-valued dual which can be used to study the invasion of fitter types after rare mutation. This is carried out in \cite{DGsel} and \cite{DGInvasion}.

math.PR↗

arXiv2011

Invasion by rare mutants in a spatial two-type Fisher-Wright system with selection

We consider a meanfield system of interacting Fisher-Wright diffusions with selection and rare mutation on the geographic space $\{1,2,...,N\}$. The type 1 has fitness 0, type 2 has fitness 1 and (rare) mutation occurs from type 1 to 2 at rate $m...N^{-1}$, selection is at rate $s>0$. The system starts in the state concentrated on type 1, the state of low fitness. We investigate this system for $N \to \infty$ on the original and large time scales. We show that for some $α\in (0,s)$ at times $α^{-1} \log N+t, t \in \R, N \to \infty$ the emergence of type 2 (positive global type-2 intensity) at a global level occurs, while at times $α^{-1} \log N+t_N$, with $t_N \to \infty$ we get fixation on type 2 and on the other hand with $t_N \to -\infty$ as $N \to \infty$ asymptotically only type 1 is present. We describe the transition from emergence to fixation in the time scale $α^{-1} \log N+t, t \in \R$ in the limit $N \to \infty$ by a McKean-Vlasov random entrance law. This entrance law behaves for $t \to -\infty$ like $^\ast\CW e^{-α|t|}$ for a positive random variable $^\ast \CW$. The formation of small droplets of type-2 dominated sites in times $o(\log N)$, or $γ...\log N, γ\in (0,α^{-1})$ is described in the limit $N \to \infty$ by a measure-valued process following a stochastic equation driven by Poissonian type noise which we identify explicitly. The total mass of this limiting $(N \to \infty)$ droplet process grows like $\CW^\ast e^{αt}$ as $t \to \infty$. We prove that exit behaviour from the small time scale equals the entrance behaviour in the large time scale, namely $\CL[^\ast\CW] =\CL[\CW^\ast]$.

math.PR↗

arXiv2011

Marked metric measure spaces

A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I. It arises as state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm- spaces and identify a convergence determining algebra of functions, called polynomials.

math.PR↗

arXiv2010

Multiscale analysis: Fisher-Wright diffusions with rare mutations and selection, logistic branching system

We study two types of stochastic processes, a mean-field spatial system of interacting Fisher-Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and a (mean-field) spatial system of supercritical branching random walks with an additional deathrate which is quadratic in the local number of particles. The former describes a standard two-type population under selection, mutation, the latter models describe a population under scarce resources causing additional death at high local population intensity. Geographic space is modelled by $\{1, \cdots, N\}$. The first process starts in an initial state with only the inferior type present or an exchangeable configuration and the second one with a single initial particle. {This material is a special case of the theory developed in \cite{DGsel}.} We study the behaviour in two time windows, first between time 0 and $T$ and secondly after a large time when in the Fisher-Wright model the rare mutants succeed respectively in the branching random walk the particle population reaches a positive spatial intensity. It is shown that the second phase for both models sets in after time $α^{-1} \log N$, if $N$ is the size of geographic space and $N^{-1}$ the rare mutation rate and $α\in (0, \infty)$ depends on the other parameters. We identify the limit dynamics in both time windows and for both models as a nonlinear Markov dynamic (McKean-Vlasov dynamic) respectively a corresponding random entrance law from time $-\infty$ of this dynamic. Finally we explain that the two processes are just two sides of the very same coin, a fact arising from duality, in particular the particle model generates the genealogy of the Fisher-Wright diffusions with selection and mutation. We discuss the extension of this duality relation to a multitype model with more than two types.

math.PR↗

arXiv2009

Coalescent processes arising in a study of diffusive clustering

This paper studies the spatial coalescent on $\Z^2$. In our setting, the partition elements are located at the sites of $\Z^2$ and undergo local delayed coalescence and migration. That is, pairs of partition elements located at the same site coalesce into one partition element after exponential waiting times. In addition, the partition elements perform independent random walks. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. These two situations are relevant if the coalescent is used to study the scaling limits for genealogies in Moran models respectively interacting Fisher-Wright diffusions (or Fleming-Viot processes), which is the key application of the present work. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box $[-t^{α/2}, t^{α/2}]^2 \cap \Z^2$ and observed at time $t^β$ with $1 \geq β\geq α\ge 0$. We study both asymptotics, as $t\to\infty$, for a fixed value of $α$ as the parameter $β\in[α,1]$ varies, and for a fixed $β$, as the parameter $α\in [0,β]$ varies. This exhibits the genealogical structure of the mono-type clusters arising in 2-dimensional Moran and Fisher-Wright systems. (... for more see the actual preprint)

math.PR↗

arXiv2009

Genealogy of catalytic branching models

We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate proportional to the number of catalyst individuals alive. The reactant forms a process in random medium. We describe asymptotically the genealogy of catalytic branching populations coded as the induced forest of $\mathbb{R}$-trees using the many individuals--rapid branching continuum limit. The limiting continuum genealogical forests are then studied in detail from both the quenched and annealed points of view. The result is obtained by constructing a contour process and analyzing the appropriately rescaled version and its limit. The genealogy of the limiting forest is described by a point process. We compare geometric properties and statistics of the reactant limit forest with those of the "classical" forest.

math.PR↗

arXiv2008

Convergence in distribution of random metric measure spaces: (Lambda-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov's idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows - provided the sequence is tight - from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Lambda-coalescents. We show that the Lambda-coalescent defines an infinite (random) metric measure space if and only if the so-called "dust-free"-property holds.

math.PR↗

arXiv2007

The renormalization transformation for two-type branching models

This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space-time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well-defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space-time scaling.

math.PR↗