Andreas Greven | Friedrich-Alexander-Universität Erlangen-Nürnberg (original) (raw)

Papers by Andreas Greven

Electronic Journal of Probability

We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals live in ... more 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 G endowed with the discrete topology. The key example of interest is the Euclidean lattice G = Z d , d ∈ N. 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. The extension in (3) allows us to place individuals in different colony when they become dormant. For each of the three models we show that the system of continuum stochastic differential equations, describing the population in the large-colony-size limit, has a unique strong solution. We also 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 seed-bank slows down the loss of genetic diversity. In model (1), the dichotomy between clustering and coexistence is determined by migration only. In particular, clustering occurs for recurrent migration and coexistence occurs for transient migration, as for the system without seed-bank. In models (2) and (3), an interesting interplay between migration and seed-bank occurs. In particular, the dichotomy is affected by the seed-bank when the wake-up time has infinite mean. For instance, for critically recurrent migration the system exhibits clustering for finite mean wake-up time and coexistence for infinite mean wake-up time. Hence, at the critical dimension for the system without seed-bank, new universality classes appear when the seed-bank is added. If the wake-up time has a sufficiently fat tail, then the seed-bank determines the dichotomy and migration has no effect at all. The presence of the seed-bank makes the proof of convergence to a unique equilibrium a conceptually delicate issue. By combining duality arguments with coupling techniques, we show that our results also hold when we replace the Fisher-Wright diffusion function by a more general diffusion function, drawn from an appropriate class.

arXiv (Cornell University), Sep 20, 2022

This is the third in a series of four papers in which we consider a system of interacting Fisher-... more This is the third in a series of four papers in which we consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type ♥ or ♦, 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 G endowed with the discrete topology. Our goal is to understand in what way the seed-bank enhances genetic diversity and causes new phenomena. In [GHO22b] we showed that the system of continuum stochastic differential equations, describing the population in the large-colony-size limit, has a unique strong solution. We further showed that if the system starts from an initial law that is invariant and ergodic under translations with a density of ♥ that is equal to θ, then it converges to an equilibrium ν θ whose density of ♥ also is equal to θ. Moreover, ν θ exhibits a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally monotype 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. Since the finite system exhibits clustering, we focus on the regime where the infinite system exhibits coexistence, which consists of two sub-regimes. If the wake-up time has finite mean, then the scaling time turns out to be proportional to the volume of the truncated geographic space, and there is a single universality class for the scaling limit, namely, the system moves through a succession of equilibria of the infinite system with a density of ♥ that evolves according to a renormalised Fisher-Wright diffusion and ultimately gets trapped in either 0 or 1. On the other hand, if the wake-up time has infinite mean, then the scaling time turns out to grow faster than the volume of the truncated geographic space, and 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. For slow growth the scaling limit is the same as when the wake-up time has finite mean, while for fast growth the scaling limit is different, namely, the density of ♥ initially remains fixed at θ, afterwards makes random switches between 0 and 1 on a range of different time scales, driven by individuals in deep seedbanks that wake up, until it finally gets trapped in either 0 or 1 on the time scale where the individuals in the deepest seed-banks wake up. Thus, the system evolves through a sequence of partial clusterings (or partial fixations) before it reaches complete clustering (or complete fixation).

arXiv (Cornell University), Apr 3, 2019

We consider the evolution of the genealogy of the population currently alive in a Feller branchin... more We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion. In contrast to the approach via labeled trees in the continuum random tree world [Ald91a, LG93], following [GPW13], the genealogies are modelled as elements of a Polish space Í which consists of all equivalence classes of ultrametric measure spaces. This space equipped with an operation called concatenation, denoted by (Í, ⊔) has a rich algebraic (semigroup) structure, [GGR19, GRG21], which is used effectively to study branching processes. We focus on the evolution of the genealogy in time and the large time asymptotics conditioned 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, such as logistic branching or state dependent branching. Furthermore the approach we take carries over very smoothly to spatial models with infinitely many components. We prove existence, uniqueness, continuity of paths and a generalized Feller property of solutions of the martingale problem for this genealogy-valued, i.e. Í-valued Feller diffusion. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. By conditioning on the entire population size process and then observing the genealogy part we obtain the precise relation to a specific time-inhomogeneous Í 1-valued Fleming-Viot process with varying resampling rate, Í 1 being the set of all equivalence classes of ultrametric probability measure spaces. This relation gives the so-called skew martingale representation of the Í-valued Feller diffusion. Via the Feynman-Kac duality we deduce the generalized branching property of the Í-valued Feller diffusion. Using a semigroup operation through concatenations on Í, [GGR19], together with the generalized branching property, [GRG21], we obtain a Lévy-Khintchine formula for the Í-valued Feller diffusion and determine explicitly the Lévy measure which has a special form, allowing us to obtain for h > 0 a decomposition into depth-h subfamilies which leads to a representation in terms of a Cox point process of genealogies where "points" correspond to single ancestor subfamilies. We determine the Í-valued process conditioned to survive until a finite time T correcting a result from the Ê +-valued literature in the computation of the diffusion coefficient. This is the key ingredient of the excursion law of the Í-valued Feller diffusion. Next we study asymptotics of the Í-valued Feller diffusion conditioned to survive forever and obtain its Kolmogorov-Yaglom limit and show that the limiting processes solve well-posed Í-valued martingale problems. Using infinite divisibility and skew martingale problems we obtain various representations of the long time limits: Í-valued 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 lineà la Evans [Eva93]. On the level of Í-valued 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 key tools for analyzing genealogies in spatial branching populations. We construct the genealogy of the interacting Feller diffusion on a countable group (super random walk) and obtain results on a martingale problem characterization, duality, generalized branching property and the long time behavior for this object. As an application we give a two scale analysis of the super random walk genealogy with strongly recurrent migration providing the asymptotic genealogy of clusters via the Í Ê-valued version of the Dawson-Watanabe process. We indicate the situation in other dimensions. Finally we enrich the Í-valued Feller process further, encoding the information on the whole population ever alive before the present time t and describe its evolution. This leads to the so called fossil process and we relate its limit for t → ∞ to the continuum random tree.

arXiv (Cornell University), Sep 9, 2012

The idea for this paper arose from discussions with P. Pfaffelhuber and A. Wakolbinger dur-

arXiv: Probability, 2015

We consider a spatial multi-type branching model in which individuals migrate in geographic space... more 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 equat...

We consider a system of interacting diffusions labeled by a geographic space that is given by the... more We consider a system of interacting diffusions labeled by a geographic space that is given by the hierarchical group OmegaN\Omega_NOmegaN of order NinmathbbNN\in\mathbb{N}NinmathbbN. 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 NinmathbbNN\in\mathbb{N}NinmathbbN, the system exhibits a dichotomy between c...

Latin American Journal of Probability and Mathematical Statistics, 2021

The paper has four goals. First, we want to generalize the classical concept of the branching pro... more 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 mathbbU\mathbb{U}mathbbU resp. mathbbUV\mathbb{U}^VmathbbUV). 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 mathbbU\mathbb{U}mathbbU-valued Feller diffusion respectively mathbbUV\mathbb{U}^VmathbbUV-valued super random walk and the historical process on countable geographic spaces the latter as two examples of...

arXiv: Probability, 2019

We consider the evolution of the genealogy of the population currently alive in a Feller branchin... more 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 mathbbU\mathbb{U}mathbbU. 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., mathbbU\mathbb{U}mathbbU-valued Feller diffusion. We give the precise relation to the time-inhomogeneous mathbbU_1\mathbb{U}_1mathbbU_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 mathbbU\mathbb{U}mathbbU, called concatenation, together with the...

Electronic Journal of Probability, 2019

The goal of this work is to decompose random populations with a genealogy in subfamilies of a giv... more 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.

Electronic Journal of Probability, 2016

For a beneficial allele which enters a large unstructured population and eventually goes to fixat... more 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 → ∞. If µ is of order α, the allele fixes (as in the spatially unstructured case) in time ∼ 2 log(α)/α. If µ is of order α γ , 0 ≤ γ ≤ 1, the fixation time is ∼ (2 + (1 − γ)∆) log(α)/α, where ∆ 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 ∼ (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.

Electronic Journal of Probability, 2016

We study the evolution of genealogies of a population of individuals, whose type frequencies resu... more 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 [DGP11]) from the case of probability measures to measures that are finite on bounded sets.

Journal of Statistical Physics, 1991

In this second part of a two-part presentation, we continue with the model introduced in Part I. ... more In this second part of a two-part presentation, we continue with the model introduced in Part I. In this part, the initial configuration has one particle at each site to the left of 0 and no particle elsewhere. The expected number of particles observed at a site moving at speed z >/0 has an exponential growth rate (speed-r growth rate) that is computed explicitly. The result reveals two characteristic wavefront speeds: %, the speed of the front of zero growth (rightmost particle), and %, the speed of the front of maximal growth. The latter speed exhibits a phase transition, changing from zero to positive as the drift in the migration crosses a threshold. The qualitative shape of the growth rate as a function of 9 changes as well. In particular, below the threshold there appears a linear piece, which corresponds to the system exhibiting an intermittency effect.

Electronic Journal of Probability, 2013

We consider the tree-valued Fleming-Viot process, (Xt) t≥0 , with mutation and selection as studi... more We consider the tree-valued Fleming-Viot process, (Xt) t≥0 , with mutation and selection as studied in Depperschmidt, Greven and 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 ε apart in the limit ε → 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 Xt 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.

Electronic Communications in Probability, 2011

A marked metric measure space (mmm-space) is a triple (X , r, µ), where (X , r) is a complete and... more A marked metric measure space (mmm-space) is a triple (X , r, µ), where (X , r) is a complete and separable metric space and µ is a probability measure on X × I 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 a 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.

Canadian Journal of Mathematics, 1995

This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a ... more This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.

When we cut an i.i.d. sequence of letters into words according to an independent renewal process,... more 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 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 quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an 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 ...

Transactions of the American Mathematical Society, 1995

Probability Theory and Related Fields, 1991

Lecture Notes in Mathematics, 2013

Electronic Journal of Probability

We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals live in ... more 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 G endowed with the discrete topology. The key example of interest is the Euclidean lattice G = Z d , d ∈ N. 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. The extension in (3) allows us to place individuals in different colony when they become dormant. For each of the three models we show that the system of continuum stochastic differential equations, describing the population in the large-colony-size limit, has a unique strong solution. We also 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 seed-bank slows down the loss of genetic diversity. In model (1), the dichotomy between clustering and coexistence is determined by migration only. In particular, clustering occurs for recurrent migration and coexistence occurs for transient migration, as for the system without seed-bank. In models (2) and (3), an interesting interplay between migration and seed-bank occurs. In particular, the dichotomy is affected by the seed-bank when the wake-up time has infinite mean. For instance, for critically recurrent migration the system exhibits clustering for finite mean wake-up time and coexistence for infinite mean wake-up time. Hence, at the critical dimension for the system without seed-bank, new universality classes appear when the seed-bank is added. If the wake-up time has a sufficiently fat tail, then the seed-bank determines the dichotomy and migration has no effect at all. The presence of the seed-bank makes the proof of convergence to a unique equilibrium a conceptually delicate issue. By combining duality arguments with coupling techniques, we show that our results also hold when we replace the Fisher-Wright diffusion function by a more general diffusion function, drawn from an appropriate class.

arXiv (Cornell University), Sep 20, 2022

This is the third in a series of four papers in which we consider a system of interacting Fisher-... more This is the third in a series of four papers in which we consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type ♥ or ♦, 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 G endowed with the discrete topology. Our goal is to understand in what way the seed-bank enhances genetic diversity and causes new phenomena. In [GHO22b] we showed that the system of continuum stochastic differential equations, describing the population in the large-colony-size limit, has a unique strong solution. We further showed that if the system starts from an initial law that is invariant and ergodic under translations with a density of ♥ that is equal to θ, then it converges to an equilibrium ν θ whose density of ♥ also is equal to θ. Moreover, ν θ exhibits a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally monotype 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. Since the finite system exhibits clustering, we focus on the regime where the infinite system exhibits coexistence, which consists of two sub-regimes. If the wake-up time has finite mean, then the scaling time turns out to be proportional to the volume of the truncated geographic space, and there is a single universality class for the scaling limit, namely, the system moves through a succession of equilibria of the infinite system with a density of ♥ that evolves according to a renormalised Fisher-Wright diffusion and ultimately gets trapped in either 0 or 1. On the other hand, if the wake-up time has infinite mean, then the scaling time turns out to grow faster than the volume of the truncated geographic space, and 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. For slow growth the scaling limit is the same as when the wake-up time has finite mean, while for fast growth the scaling limit is different, namely, the density of ♥ initially remains fixed at θ, afterwards makes random switches between 0 and 1 on a range of different time scales, driven by individuals in deep seedbanks that wake up, until it finally gets trapped in either 0 or 1 on the time scale where the individuals in the deepest seed-banks wake up. Thus, the system evolves through a sequence of partial clusterings (or partial fixations) before it reaches complete clustering (or complete fixation).

arXiv (Cornell University), Apr 3, 2019

We consider the evolution of the genealogy of the population currently alive in a Feller branchin... more We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion. In contrast to the approach via labeled trees in the continuum random tree world [Ald91a, LG93], following [GPW13], the genealogies are modelled as elements of a Polish space Í which consists of all equivalence classes of ultrametric measure spaces. This space equipped with an operation called concatenation, denoted by (Í, ⊔) has a rich algebraic (semigroup) structure, [GGR19, GRG21], which is used effectively to study branching processes. We focus on the evolution of the genealogy in time and the large time asymptotics conditioned 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, such as logistic branching or state dependent branching. Furthermore the approach we take carries over very smoothly to spatial models with infinitely many components. We prove existence, uniqueness, continuity of paths and a generalized Feller property of solutions of the martingale problem for this genealogy-valued, i.e. Í-valued Feller diffusion. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. By conditioning on the entire population size process and then observing the genealogy part we obtain the precise relation to a specific time-inhomogeneous Í 1-valued Fleming-Viot process with varying resampling rate, Í 1 being the set of all equivalence classes of ultrametric probability measure spaces. This relation gives the so-called skew martingale representation of the Í-valued Feller diffusion. Via the Feynman-Kac duality we deduce the generalized branching property of the Í-valued Feller diffusion. Using a semigroup operation through concatenations on Í, [GGR19], together with the generalized branching property, [GRG21], we obtain a Lévy-Khintchine formula for the Í-valued Feller diffusion and determine explicitly the Lévy measure which has a special form, allowing us to obtain for h > 0 a decomposition into depth-h subfamilies which leads to a representation in terms of a Cox point process of genealogies where "points" correspond to single ancestor subfamilies. We determine the Í-valued process conditioned to survive until a finite time T correcting a result from the Ê +-valued literature in the computation of the diffusion coefficient. This is the key ingredient of the excursion law of the Í-valued Feller diffusion. Next we study asymptotics of the Í-valued Feller diffusion conditioned to survive forever and obtain its Kolmogorov-Yaglom limit and show that the limiting processes solve well-posed Í-valued martingale problems. Using infinite divisibility and skew martingale problems we obtain various representations of the long time limits: Í-valued 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 lineà la Evans [Eva93]. On the level of Í-valued 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 key tools for analyzing genealogies in spatial branching populations. We construct the genealogy of the interacting Feller diffusion on a countable group (super random walk) and obtain results on a martingale problem characterization, duality, generalized branching property and the long time behavior for this object. As an application we give a two scale analysis of the super random walk genealogy with strongly recurrent migration providing the asymptotic genealogy of clusters via the Í Ê-valued version of the Dawson-Watanabe process. We indicate the situation in other dimensions. Finally we enrich the Í-valued Feller process further, encoding the information on the whole population ever alive before the present time t and describe its evolution. This leads to the so called fossil process and we relate its limit for t → ∞ to the continuum random tree.

arXiv (Cornell University), Sep 9, 2012

The idea for this paper arose from discussions with P. Pfaffelhuber and A. Wakolbinger dur-

arXiv: Probability, 2015

We consider a spatial multi-type branching model in which individuals migrate in geographic space... more 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 equat...

We consider a system of interacting diffusions labeled by a geographic space that is given by the... more We consider a system of interacting diffusions labeled by a geographic space that is given by the hierarchical group OmegaN\Omega_NOmegaN of order NinmathbbNN\in\mathbb{N}NinmathbbN. 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 NinmathbbNN\in\mathbb{N}NinmathbbN, the system exhibits a dichotomy between c...

Latin American Journal of Probability and Mathematical Statistics, 2021

The paper has four goals. First, we want to generalize the classical concept of the branching pro... more 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 mathbbU\mathbb{U}mathbbU resp. mathbbUV\mathbb{U}^VmathbbUV). 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 mathbbU\mathbb{U}mathbbU-valued Feller diffusion respectively mathbbUV\mathbb{U}^VmathbbUV-valued super random walk and the historical process on countable geographic spaces the latter as two examples of...

arXiv: Probability, 2019

We consider the evolution of the genealogy of the population currently alive in a Feller branchin... more 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 mathbbU\mathbb{U}mathbbU. 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., mathbbU\mathbb{U}mathbbU-valued Feller diffusion. We give the precise relation to the time-inhomogeneous mathbbU_1\mathbb{U}_1mathbbU_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 mathbbU\mathbb{U}mathbbU, called concatenation, together with the...

Electronic Journal of Probability, 2019

The goal of this work is to decompose random populations with a genealogy in subfamilies of a giv... more 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.

Electronic Journal of Probability, 2016

For a beneficial allele which enters a large unstructured population and eventually goes to fixat... more 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 → ∞. If µ is of order α, the allele fixes (as in the spatially unstructured case) in time ∼ 2 log(α)/α. If µ is of order α γ , 0 ≤ γ ≤ 1, the fixation time is ∼ (2 + (1 − γ)∆) log(α)/α, where ∆ 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 ∼ (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.

Electronic Journal of Probability, 2016

We study the evolution of genealogies of a population of individuals, whose type frequencies resu... more 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 [DGP11]) from the case of probability measures to measures that are finite on bounded sets.

Journal of Statistical Physics, 1991

In this second part of a two-part presentation, we continue with the model introduced in Part I. ... more In this second part of a two-part presentation, we continue with the model introduced in Part I. In this part, the initial configuration has one particle at each site to the left of 0 and no particle elsewhere. The expected number of particles observed at a site moving at speed z >/0 has an exponential growth rate (speed-r growth rate) that is computed explicitly. The result reveals two characteristic wavefront speeds: %, the speed of the front of zero growth (rightmost particle), and %, the speed of the front of maximal growth. The latter speed exhibits a phase transition, changing from zero to positive as the drift in the migration crosses a threshold. The qualitative shape of the growth rate as a function of 9 changes as well. In particular, below the threshold there appears a linear piece, which corresponds to the system exhibiting an intermittency effect.

Electronic Journal of Probability, 2013

We consider the tree-valued Fleming-Viot process, (Xt) t≥0 , with mutation and selection as studi... more We consider the tree-valued Fleming-Viot process, (Xt) t≥0 , with mutation and selection as studied in Depperschmidt, Greven and 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 ε apart in the limit ε → 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 Xt 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.

Electronic Communications in Probability, 2011

A marked metric measure space (mmm-space) is a triple (X , r, µ), where (X , r) is a complete and... more A marked metric measure space (mmm-space) is a triple (X , r, µ), where (X , r) is a complete and separable metric space and µ is a probability measure on X × I 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 a 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.

Canadian Journal of Mathematics, 1995

This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a ... more This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.

When we cut an i.i.d. sequence of letters into words according to an independent renewal process,... more 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 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 quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an 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 ...

Transactions of the American Mathematical Society, 1995

Probability Theory and Related Fields, 1991

Lecture Notes in Mathematics, 2013