arXiv subjects

Andreas Greven

Publications and source records attributed to Andreas Greven.

At least 19 recordsLinked to original sources

arXiv2023

Continuum graph dynamics via population dynamics: well-posedness, duality and equilibria

This paper introduces graphemes for constructing and analyzing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which capture the static properties of dense graphs via exchangeability or subgraph densities, graphemes are capable of modeling the full space-time evolution of graphs. A grapheme is an equivalence class of triples: (Polish space, symmetric {0,1}-valued connection function, sampling probability measure). We focus on embeddings in ultrametric spaces, encoding the graph history and linking directly to population dynamics models. Graphemes utilize stronger equivalences (homeomorphism, isometry) than graphons. We construct grapheme-valued Markov processes as limits of finite graph evolutions, driven by Fleming-Viot, Dawson-Watanabe, and McKean-Vlasov analogues. We establish characterization via well-posed martingale problems, yielding strong Markov processes with the Feller property and continuous paths (diffusions). Duality relations involving coalescent processes are derived. We identify non-trivial equilibria, linked to classical distributions from population genetics. This framework extends [arXiv:1908.06241] by incorporating history, enabling rigorous analysis via martingale problems, and characterizing non-trivial long-term behavior.

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arXiv2023

The grapheme-valued Wright-Fisher diffusion with mutation

In [Athreya, den Hollander, Röllin; 2021, arXiv:1908.06241] models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterised as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set, an adjacency matrix and a sampling measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Poisson-Dirichlet distribution. In a companion paper [Greven, den Hollander, Klimovsky, Winter; 2023], we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.

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arXiv2023

Duality and the well-posedness of a martingale problem

For two Polish state spaces $E_X$ and $E_Y$, and an operator $G_X$, we obtain existence and uniqueness of a $G_X$-martingale problem provided there is a bounded continuous duality function $H$ on $E_X \times E_Y$ together with a dual process $Y$ on $E_Y$ which is the unique solution of a $G_Y$-martingale problem. For the corresponding solutions $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$, duality with respect to a function $H$ in its simplest form means that the relation $\mathbb E_x[H(X_t,y)] = \mathbb E_y[H(x,Y_t)]$ holds for all $(x,y) \in E_X \times E_Y$ and $t\ge 0$. While duality is well-known to imply uniqueness of the $G_X$-martingale problem, we give here a set of conditions under which duality also implies existence without using approximating sequences of processes of a different kind (e.g.\ jump processes to approximate diffusions) which is a widespread strategy for proving existence of solutions of martingale problems. Given the process $(Y_t)_{t\ge 0}$ and a duality function $H$, to prove existence of $(X_t)_{t\ge 0}$ one has to show that the r.h.s.\ of the duality relation defines for each $y$ a measure on $E_X$, i.e.\ there are transition kernels $(μ_t)_{t\geq 0}$ from $E_X$ to $E_X$ such that $\mathbb E_y[H(x,Y_t)] = \int μ_t(x,dx')\, H(x',y)$ for all $(x,y) \in E_X \times E_Y$ and all $t\geq 0$. As examples, we treat resampling and branching models, such as the Fleming-Viot measure-valued diffusion and its spatial counterparts (with both, discrete and continuum space), as well as branching systems, such as Feller's branching diffusion. While our main result as well as all examples come with (locally) compact state spaces, we discuss the strategy to lift our results to genealogy-valued processes or historical processes, leading to non-compact (discrete and continuum) state spaces. Such applications will be tackled in forthcoming work based on the present article.

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arXiv2022

Quenched large deviation principle for words in a letter sequence

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the \emph{annealed} large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the \emph{quenched} LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an \emph{algebraic} tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. The previous version (arXiv:0807.2611v2) appeared in Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403--456. Meanwhile, it has turned out that the original proof of the representation of the rate function is flawed when the mean word length is infinite. We add an erratum in which we fix the flaw in the proof. Along the way we derive new representations of the rate function that are interesting in their own right. A key ingredient in the proof is the observation that if the rate function in the annealed large deviation principle is finite at a stationary word process, then the letters in the tail of the long words in this process are typical.

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arXiv2022

Spatial populations with seed-bank: finite-systems scheme

We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type one of two types, live in colonies, and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labelling the colonies we consider a countable Abelian group endowed with the discrete topology. In earlier work we showed that the system has a one-parameter family of equilibria controlled by the relative density of the two types. Moreover, these equilibria exhibit a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium). We identified the parameter regimes for which these two phases occur, and found that these regimes are different when the mean wake-up time of a dormant individual is finite or infinite. The goal of the present paper is to establish the finite-systems scheme, i.e., identify how a finite truncation of the system (both in the geographic space and in the seed-bank) behaves as both the time and the truncation level tend to infinity, properly tuned together. If the wake-up time has finite mean, then there is a single universality class for the scaling limit. On the other hand, if the wake-up time has infinite mean, then there are two universality classes depending on how fast the truncation level of the seed-bank grows compared to the truncation level of the geographic space.

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arXiv2021

Spatial Populations with seed-bank: renormalisation on the hierarchical group

We consider a system of interacting diffusions labeled by a geographic space that is given by the hierarchical group $Ω_N$ of order $N\in\mathbb{N}$. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. The migration kernel has a hierarchical structure: individuals hop between colonies at a rate that depends on the hierarchical distance between the colonies. The seed-bank has a layered structure: when individuals become dormant they acquire a colour that determines the rate at which they become active again. The latter allows us to model seed-banks whose wake-up times have a fat tail. We analyse a system of coupled stochastic differential equations that describes the population in the large-colony-size limit. For fixed $N\in\mathbb{N}$, the system exhibits a dichotomy between coexistence and clustering. We identify the range of parameters controlling the migration and the seed-bank for which clustering prevails. We carry out a multi-scale renormalisation analysis in the hierarchical mean-field limit $N\to\infty$. We show that block averages on hierarchical space-time scale $k \in \mathbb{N}$ perform a diffusion with a renormalised diffusion function that depends on $k$. In the clustering regime, after an appropriate scaling with $k$, this diffusion function converges to the Fisher-Wright diffusion function as $k\to\infty$, irrespective of the diffusion function controlling the resampling. For several subclasses of parameters we identify the speed at which the scaled renormalised diffusion function converges to the Fisher-Wright diffusion as $k\to\infty$. We show that the seed-bank reduces the speed compared to the model without seed-bank.

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arXiv2020

Branching Processes -- A General Concept

The paper has four goals. First, we want to generalize the classical concept of the branching property so that it becomes applicable for historical and genealogical processes (using the coding of genealogies by ($V$-marked) ultrametric measure spaces leading to state spaces $\mathbb{U}$ resp. $\mathbb{U}^V$). The processes are defined by well-posed martingale problems. In particular we want to complement the corresponding concept of infinite divisibility developed in \cite{infdiv} for this context. Second one of the two main points, we want to find a corresponding characterization of the generators of branching processes more precisely their martingale problems which is both easy to apply and general enough to cover a wide range of state spaces. As a third goal we want to obtain the branching property of the $\mathbb{U}$-valued Feller diffusion respectively $\mathbb{U}^V$-valued super random walk and the historical process on countable geographic spaces the latter as two examples of a whole zoo of spatial processes we could treat. The fourth goal is to show the robustness of the method and to get the branching property for genealogies marked with ancestral path, giving the line of descent moving through the ancestors and space, leading to path-marked ultra-metric measure spaces. This processes are constructed here giving our second major result. The starting point for all four points is the Feller diffusion model, the final goal the (historical) super random walk model.

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arXiv2020

Spatial populations with seed-bank: well-posedness, duality and equilibrium

We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labelling the colonies we consider a countable Abelian group $\mathbb{G}$ endowed with the discrete topology. The key example of interest is the Euclidean lattice $\mathbb{G}=\mathbb{Z}^d$. Our goal is to classify the long-time behaviour of the system in terms of the underlying model parameters. In particular, we want to understand in what way the seed-bank enhances genetic diversity. We introduce three models of increasing generality, namely, individuals become dormant: (1) in the seed-bank of their colony; (2) in the seed-bank of their colony while adopting a random colour that determines their wake-up time; (3) in the seed-bank of a random colony while adopting a random colour. The extension in (2) allows us to model wake-up times with fat tails while preserving the Markov property of the evolution. For each of the three models we show that the system converges to a unique equilibrium depending on a single density parameter that is determined by the initial state, and exhibits a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium) depending on the parameters controlling the migration and the seed-bank. The dichotomy between clustering and coexistence in model 1 is determined by migration only. In models (2) and (3), when the wake-up time has infinite mean, the dichotomy is determined by both the exchange with the seed-bank and migration. It turns out that the seed-bank affects the long-time behaviour both quantitatively and qualitatively.

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arXiv2019

Stochastic Evolution of spatial populations: From configurations to genealogies and back

The paper reviews the results obtained for spatial population models and the evolution of the genealogies of these populations during the last decade by the author and his coworkers. The focus is on their large scale behaviour and on the analysis of universality classes of large scale behaviour via the methods of the hierarchical mean-field limit or via the spatial continuum limit and from another angel, the finite system scheme . We use genealogical information to analyze the type and location structure and vice versa. To apply this approach and to explain effects in biological situations we extend the classical model classes in new directions. Namely we look as population models here at: Fleming-Viot genealogies in continuum geographic space, Cannings models with block resampling (reducing diversity), Fisher-Wright diffusion with coloured seedbanks (enhancing diversity) and evolving genealogies of Fleming-Viot models with selection and rare mutation.

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arXiv2019

Branching trees I: Concatenation and infinite divisibility

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 collections of subfamilies of fixed kinship h (described as ultrametric measure spaces), for every depth h as a measurable functional of the genealogy. 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 Levy-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 [DG18b] 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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arXiv2019

Genealogy-valued Feller diffusion

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, the elements of the space $\mathbb{U}$. 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 prove existence, uniqueness and Feller property of solutions of the martingale problem for this genealogy valued, i.e., $\mathbb{U}$-valued Feller diffusion. We give the precise relation to the time-inhomogeneous $\mathbb{U}_1$-valued Fleming-Viot process. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. Using a semigroup operation on $\mathbb{U}$, called concatenation, together with the branching property we obtain a L\'evy-Khintchine formula for $\mathbb{U}$-valued Feller diffusion and we determine explicitly the L\'evy measure on $\mathbb{U}\setminus\{0\}$. From this we obtain for $h>0$ the decomposition into depth-$h$ subfamilies, a representation of the process as concatenation of a Cox point process of genealogies of single ancestor subfamilies. Furthermore, we will identify the $\mathbb{U}$-valued process conditioned to survive until a finite time $T$. We study long time asymptotics, such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the level of ultrametric measure spaces. We also obtain various representations of the long time limits.

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arXiv2018

Stochastic evolution of genealogies of spatial populations: state description, characterization of dynamics and properties

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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arXiv2017

The hierarchical Cannings process in random environment

In an earlier paper, we introduced and studied a system of hierarchically interacting measure-valued random processes which describes a large population of individuals carrying types and living in colonies labelled by the hierarchical group of order $N$. The individuals are subject to migration, resampling on all hierarchical scales simultaneously. Upon resampling, a random positive fraction of the population in a block of colonies inherits the type of a random single individual in that block, which is why we refer to our system as the hierarchical Cannings process. In the present paper, we study a version of the hierarchical Cannings process in random environment, namely, the resampling measures controlling the change of type of individuals in different blocks are chosen randomly with a given mean and are kept fixed in time (= the quenched setting). We give a necessary and sufficient condition under which a multi-type equilibrium is approached (= coexistence) as opposed to a mono-type equilibrium (= clustering). Moreover, in the hierarchical mean-field limit $N \to \infty$, with the help of a renormalization analysis, we obtain a full picture of the space-time scaling behaviour of block averages on all hierarchical scales simultaneously. We show that the $k$-block averages are distributed as the superposition of a Fleming-Viot diffusion with a deterministic volatility constant $d_k$ and a Cannings process with a random jump rate, both depending on $k$. In the random environment, $d_k$ turns out to be smaller than in the homogeneous environment of the same mean. We investigate how $d_k$ scales with $k$. This leads to five universality classes of cluster formation in the mono-type regime. We find that if clustering occurs, then the random environment slows down the growth of the clusters, i.e., enhances the diversity of types.

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arXiv2017

Continuum Space Limit of the Genealogies of Interacting Fleming-Viot Processes on $\Z$

We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on $\Z$. We construct and analyze the genealogical structure of the population in this genealogy-valued Fleming-Viot process as a marked metric measure space, with each individual carrying its spatial location as a mark. We then show that its time evolution converges to that of the genealogy of a continuum-sites stepping stone model on $\R$, if space and time are scaled diffusively. We construct the genealogies of the continuum-sites stepping stone model as functionals of the Brownian web, and furthermore, we show that its evolution solves a martingale problem. The generator for the continuum-sites stepping stone model has a singular feature: at each time, the resampling of genealogies only affects a set of individuals of measure $0$. Along the way, we prove some negative correlation inequalities for coalescing Brownian motions, as well as extend the theory of marked metric measure spaces (developed recently by Depperschmidt, Greven and Pfaffelhuber [DGP12]) from the case of probability measures to measures that are finite on bounded sets.

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arXiv2016

The fixation time of a strongly beneficial allele in a structured population

For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately $2\log(α)/α$ for a large selection coefficient $α$. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $μ$ for which the fixation times have different asymptotics as $α\to \infty$. If $μ$ is of order $α$, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log(α)/α$. If $μ$ is of order $α^γ, 0\leq γ\leq 1$, the fixation time is $\sim (2 + (1-γ)Δ) \log(α)/α$, where $Δ$ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If $μ= 1/\log(α)$, the fixation time is $\sim (2+S)\log(α)/α$, where $S$ is a random time in a simple epidemic model. The main idea for our analysis is to combine a new moment dual for the process conditioned to fixation with the time reversal in equilibrium of a spatial version of Neuhauser and Krone's ancestral selection graph.

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arXiv2015

Multi-type spatial branching models for local self-regulation I: Construction and an exponential duality

We consider a spatial multi-type branching model in which individuals migrate in geographic space according to random walks and reproduce according to a state-dependent branching mechanism which can be sub-, super- or critical depending on the local intensity of individuals of the different types. The model is a Lotka-Volterra type model with a spatial component and is related to two models studied in \cite{BlathEtheridgeMeredith2007} as well as to earlier work in \cite{Etheridge2004} and in \cite{NeuhauserPacala1999}. Our main focus is on the diffusion limit of small mass, locally many individuals and rapid reproduction. This system differs from spatial critical branching systems since it is not density preserving and the densities for large times do not depend on the initial distribution but mainly on the carrying capacities. We prove existence of the infinite particle model and the system of interacting diffusions as solutions of martingale problems or systems of stochastic equations. In the exchangeable case in which the parameters are not type dependent we show uniqueness of the solutions. For that purpose we establish a new exponential duality.

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arXiv2014

Renormalisation of hierarchically interacting Cannings processes

The present paper brings a new class of interacting jump processes into focus. We start from a single-colony $C^Λ$-process, which arises as the continuum-mass limit of a $Λ$-Cannings individual-based population model, where $Λ$ is a finite non-negative measure that describes the offspring mechanism. After that we introduce a system of hierarchically interacting $C^Λ$-processes, where the interaction comes from migration and reshuffling-resampling based on measures $(Λ_k)_{k}$ both acting in $k$-blocks of the hierarchical group. We refer to this system as the $C_N^{c,Λ}$-process. The dual process of the $C_N^{c,Λ}$-process is a spatial coalescent with multi-level block coalescence. For the above system we carry out a full renormalisation analysis in the hierarchical mean-field limit $N\to\infty$. Our main result is that, in the limit as $N\to\infty$, on each scale $k\in\mathbb{N}_0$ the $k$-block averages of the $C_N^{c,Λ}$-process converge to a random process that is a superposition of a $C^{Λ_k}$-process and a Fleming-Viot process, the latter with a volatility $d_k$ and with a drift of strength $c_k$ towards the limiting $(k+1)$-block average. It turns out that $d_k$ is a function of $c_l$ and $Λ_l$ for all $0\leq l<k$. Thus, it is through the volatility that the renormalisation manifests itself. We discuss the implications of the scaling of $d_k$ for the behaviour on large space-time scales of the $C_N^{c,Λ}$-process. We compare the outcome with what is known from the renormalisation analysis of hierarchically interacting Fleming-Viot diffusions, pointing out several new features. We obtain a new classification for when the process exhibits clustering, respectively, exhibits local coexistence. Finally, we show that for finite $N$ the same dichotomy between clustering and local coexistence holds as for $N\to\infty$.

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arXiv2013

Path-properties of the tree-valued Fleming-Viot process

We consider the tree-valued Fleming-Viot process, $(\mathcal X_t)_{t\geq 0}$, with mutation and selection as studied in Depperschmidt, Greven, Pfaffelhuber (2012). This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming-Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent. In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the neutral Fleming-Viot genealogies at fixed time $t$ arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than $\ve$ apart in the limit $\ve\to 0$. Second, we answer two open questions about the sample paths of the tree-valued Fleming-Viot process. We show that for all $t>0$ almost surely the marked metric measure space $\mathcal X_t$ has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming-Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov's formula giving absolute continuity.

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