Path-properties of the tree-valued Fleming–Viot process (original) (raw)
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Tree-valued Fleming–Viot dynamics with mutation and selection
Annals of Applied Probability an Official Journal of the Institute of Mathematical Statistics, 2012
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.
Electronic Journal of Probability, 2016
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.
Genealogy-valued Feller diffusion
arXiv (Cornell University), 2019
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.
Limit theorems for Markov processes indexed by continuous time Galton–Watson trees
The Annals of Applied Probability, 2011
We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Lévy processes.
The spatial Lambda-Fleming-Viot process with fluctuating selection
Electronic Journal of Probability, 2021
We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our population is assumed to be composed of just two genetic types. Short bursts of selection acting in opposing directions drive to maintain both types at intermediate frequencies, while the fluctuations due to 'genetic drift' work to eliminate variation in the population. We consider first a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. As is usual with spatially distributed populations, in dimensions greater than one, the 'genetic drift' disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic p.d.e. driven by a noise that is white in time but coloured in space. We discuss the (rather limited) situations under which there is a duality with a system of branching and annihilating particles. We also write down a system of equations that captures the frequency of descendants of particular subsets of the population and use this same idea of 'tracers', which we learned from Hallatschek and Nelson (2008) and Durrett and Fan (2016), in numerical experiments with a closely related model based on the classical Moran model.
arXiv: Probability, 2019
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...
A class of infinite-dimensional diffusion processes with connection to population genetics
Journal of Applied Probability, 2007
Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite dimensional diffusion processes on ∆ ∞ := {x ∈ [0, 1] N : i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence to the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process which does not satisfy the log-Sobolev inequality when S is infinite as observed by W. Stannat .
On invariant measures for simple branching processes (Summary)
Bulletin of the Australian Mathematical Society, 1970
Galton-Watson process, whether with or without immigration, may be discussed in terms of measures of a subcritical process with a possibly defective immigration distribution. There is in fact only one such measure satisfying a regular variation condition.