Using the lookdown construction of Donnelly and Kurtz we prove that, at any fixed positive time, the $Λ$-Fleming-Viot process with underlying Brownian motion has a compact support provided that the corresponding $Λ$-coalescent comes down from infinity not too slowly. We also find both upper bound and lower bound on the Hausdorff dimension for the support.
For a probability-measure-valued neutral Fleming-Viot process $Z_t$ with Lévy mutation and resampling mechanism associated to a general $Λ$-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time $t>0$, with probability one the closed support $S(Z_t)$ of the Fleming-Viot process satisfies $S(ν* Z_t) \subseteq S(Z_t)$, where $ν$ is the Lévy measure of the mutation process. To show this result, we apply Donnelly-Kurtz's lookdown particle representation for Fleming-Viot process.
We prove under mild conditions that the Fleming-Viot process selects the minimal quasi-stationary distribution for Markov processes with soft killing on non-compact state spaces. Our results are applied to multi-dimensional birth and death processes, continuous time Galton-Watson processes and diffusion processes with soft killing.
Estimates for exit time from an interval of length 2r before a prescribed time T are derived for solutions of a class of stochastic partial differential equations used to characterize two population models: super-Brownian motion and Fleming-Viot Process. These types of estimates are then derived for the two population models. The corresponding large deviation results are also applied for the acquired bounds.
We show that a space-time rescaling of the spatial Lamba-Fleming-Viot process of Barton and Etheridge converges to super-Brownian motion. This can be viewed as an extension of a result of Chetwynd-Diggle and Etheridge (2018). In that work the scaled impact factors (which govern the event based dynamics) vanish in the limit, here we drop that requirement. The analysis is particularly interesting in the biologically relevant two-dimensional case.
A Central Limit Theorem for Fleming-Viot Particle Systems with Soft Killing
假设在给定时刻仍有生命的马尔可夫过程，其分布可以用Fleming-Viot型粒子系统来近似.在这样的系统中，每个粒子都是根据基本的马尔可夫过程的定律独立模拟的，并且在每个杀死时间分支到另一个粒子上。这种方法在大人口限制中的一致性是最近几篇文章的主题。在本文中，我们进一步证明了Fleming-Viot粒子系统在给定时刻的一个中心极限定理：一个“软杀”假设和一个包含潜在马尔可夫过程的“carré du champ”算子的有界性条件.
We study the large scale behaviour of a population consisting of two types which evolve in dimension d = 1, 2 according to a spatial Lambda- Fleming-Viot process subject to random time-independent selection. If one of the two types is rare compared to the other, we prove that its evolution can be approximated by a super-Brownian motion in a random (and singular) environment. Without the sparsity assumption, a diffusion approximation leads to a Fisher-KPP equation in a random potential. The proofs build on two-scale Schauder estimates and semidiscrete approximations of the Anderson Hamiltonian.
For a class of $Λ$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $Λ$-coalescents come down from infinity, we obtain sharp global and local modulus of continuities for the ancestry processes recovered from the lookdown representations. As applications, we prove both global and local modulus of continuities for the $Λ$-Fleming-Viot support processes. In particular, if the $Λ$-coalescent is the Beta$(2-β,β)$ coalescent for $β\in(1,2]$ with $β=2$ corresponding to Kingman's coalescent, then for $h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity hold…
A class of Fleming-Viot processes with decaying sampling rates and $α$-stable motions that correspond to distributions with growing populations are introduced and analyzed. Almost sure long-time scaling limits for these processes are developed, addressing the question of long-time population distribution for growing populations. Asymptotics in higher orders are investigated. Convergence of particle location occupation and inhabitation time processes are also addressed and related by way of the historical process. The basic results and techniques allow general Feller motion/mutation and may app…
We study the Fleming-Viot particle process formed by N interacting continuous-time asymmetric random walks on the cycle graph, with uniform killing. We show that this model has a remarkable exact solvability, despite the fact that it is non-reversible with non-explicit invariant distribution. Our main results include quantitative propagation of chaos and exponential ergodicity with explicit constants, as well as formulas for covariances at equilibrium in terms of the Chebyshev polynomials. We also obtain a bound uniform in time for the convergence of the proportion of particles in each state w…
This article presents a variant of Fleming-Viot particle systems, which are a standard way to approximate the law of a Markov process with killing as well as related quantities. Classical Fleming-Viot particle systems proceed by simulating $N$ trajectories, or particles, according to the dynamics of the underlying process, until one of them is killed. At this killing time, the particle is instantaneously branched on one of the $(N-1)$ other ones, and so on until a fixed and finite final time $T$. In our variant, we propose to wait until $K$ particles are killed and then rebranch them independe…
In previous work, we constructed Fleming--Viot-type measure-valued diffusions (and diffusions on a space of interval partitions of the unit interval $[0,1]$) that are stationary with the Poisson--Dirichlet laws with parameters $α\in(0,1)$ and $θ\geq 0$. In this paper, we complete the proof that these processes resolve a conjecture by Feng and Sun (2010) by showing that the processes of ranked atom sizes (or of ranked interval lengths) of these diffusions are members of a two-parameter family of diffusions introduced by Petrov (2009), extending a model by Ethier and Kurtz (1981) in the case $α=…
We investigate the behaviour of an establishing mutation which is subject to rapidly fluctuating selection under the Lambda-Fleming-Viot model and show that under a suitable scaling it converges to the Feller diffusion in a random environment. We then extend to a population that is distributed across a spatial continuum. In this setting the scaling limit is the SuperBrownian motion in a random environment.The scaling results for the behaviour of the rare allele are achieved via particle representations which belong to the family of `lookdown constructions'. This generalises the results obtaine…
Tree-valued Fleming-Viot dynamics with mutation and selection
Fleming-Viot测度值扩散是一个描述突变、选择和随机繁殖下（等位基因）类型进化的马尔可夫过程。我们丰富了这一过程中的系谱关系的个人，使随机型分布以及在人口的系谱距离随机演变。具有突变和选择的Fleming-Viot动力学（TFVMS）的这种树值富集的状态空间由标记超度量测度空间组成，配备有标记Gromov-弱拓扑和适当的多项式概念作为检验函数的分离代数。TFVMS的构造和研究是基于一个适定的鞅问题。对于存在性，我们使用近似的有限总体模型，树值Moran模型，而唯一性遵循从对偶到函数值过程。由于绝对连续性，所得到的过程的路径性质从中性情况下进行，由标记度量测度空间上的新Girsanov型定理给出。为了研究该过程的长时间行为，我们使用了基于Dawson和Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation（2011 c）Unpublished manuscript]思想的对偶性，并证明了TFVMS的遍历性，如果Fleming-Viot测度值扩散是遍历的。作为进一步的应用，我们考虑两个等位基因类型和加性选择的情况。对于小的选择强度，我们给出了一个扩展的拉普拉斯变换的家谱距离在平衡，这是第一步，表明距离较短的选择的情况下。
The Lambda-Fleming-Viot process and a connection with Wright-Fisher diffusion
作用于函数g（x）的d维Lambda-Fleming-Viot生成器，其中x是d个等位基因频率的向量，可以写为作用于函数g的Wright-Fisher生成器，其中x的修改的随机线性变元由Lambda-Fleming-Viot过程中发生的分割引起。特征值和右多项式特征向量很容易从这个表示看到。二维过程，其中有一个一维的发电机，被认为是详细的。格林函数的非线性方程。在一个模型与基因选择的证明，有一个关键的选择值，如果选择系数大于或等于临界值，然后固定，当边界1被击中，有概率1开始从任何非零频率。这是一个分析证明，不同于Der，Epstein and Plotkin（2011）和Foucart（2013）的证明。当注视不确定时，本文提出了一种算法来计算注视概率。在无限多等位基因Lambda-Fleming-Viot过程中的一个应用是发现基于大小偏置的等位基因频谱的有趣恒等式。Fleming-Viot过程中的矩对偶过程是时间上常见的λ-聚结树。Wright-Fisher表示采用不同的多项式g_n（x）作为检验函数，产生了一个对偶纯灭过程，它与Kingman合并过程相似，并以1为单位递减。当用g_n（x）表示时，该过程的特征值类似于雅可比多项式，扮演x^n的角色。当有突变时，平稳分布下的E[g_n（X）]类似于β分布中的n阶矩。当X和n是d维时，有一个版本g_n（X），当d趋于无穷大时，甚至…
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 proces…
