arXiv subjects

Anton Bovier

Publications and source records attributed to Anton Bovier.

At least 37 records · Page 2Linked to original sources

arXiv2017

The Polymorphic Evolution Sequence for Populations with Phenotypic Plasticity

In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth- and death rates as well as the competition kernel, $c(x,y)$ which describes the induced death rate that an individual of type $x$ experiences due to the presence of an individual or type $y$. When a new individual is born, with a small probability a mutation occurs, i.e. the offspring has different genotype as the parent. The novel aspect of the models we study is that an individual with a given genotype may express a certain set of different phenotypes, and during its lifetime it may switch between different phenotypes, with rates that are much larger then the mutation rates and that, moreover, may depend on the state of the entire population. The evolution of the population is described by a continuous-time, measure-valued Markov process. In our last paper [4], such a model was proposed to describe tumor evolution under immunotherapy. In the present paper we consider a large class of models which comprises the example studied in [4] and analyse their scaling limits as the population size tends to infinity and the mutation rate tends to zero. Under suitable assumptions, we prove convergence to a Markov jump process that is a generalisation of the polymorphic evolution sequence (PES) as analysed by Champagnat and Méléard [8, 10].

math.PR↗

arXiv2016

Extended Convergence of the Extremal Process of Branching Brownian Motion

We extend the results of Arguin et al and A\"ıdékon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_+\times \mathbb{R}$ where each cluster is the atom of a Poisson point process on $\mathbb{R}_+\times \mathbb{R}$ with a random intensity measure $Z(dz) \times Ce^{-\sqrt 2x}dx$, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous result for the Gaussian free field by Biskup and Louidor.

math.PR↗

arXiv2016

Survival of a recessive allele in a Mendelian diploid model

In this paper we analyse the genetic evolution of a diploid hermaphroditic population, which is modelled by a three-type nonlinear birth-and-death process with competition and Mendelian reproduction. In a recent paper, Collet et al., 2013 have shown that, on the mutation time-scale, the process converges to the Trait-Substitution Sequence of adaptive dynamics, stepping from one homozygotic state to another with higher fitness. We prove that, under the assumption that a dominant allele is also the fittest one, the recessive allele survives for a time of order at least K^{1/4-a}, where K is the size of the population and a>0.

math.PR↗

arXiv2016

A stochastic individual-based model for immunotherapy of cancer

We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range of biological applications. Our primary motivation is modelling of immunotherapy of malignant tumours. In this context the different actors, T-cells, cytokines or cancer cells, are modelled as single particles (individuals) in the stochastic system. The main expansions of the model are distinguishing cancer cells by phenotype and genotype, including environment-dependent phenotypic plasticity that does not affect the genotype, taking into account the effects of therapy and introducing a competition term which lowers the reproduction rate of an individual in addition to the usual term that increases its death rate. We illustrate the new setup by using it to model various phenomena arising in immunotherapy. Our aim is twofold: on the one hand, we show that the interplay of genetic mutations and phenotypic switches on different timescales as well as the occurrence of metastability phenomena raise new mathematical challenges. On the other hand, we argue why understanding purely stochastic events (which cannot be obtained with deterministic models) may help to understand the resistance of tumours to therapeutic approaches and may have non-trivial consequences on tumour treatment protocols. This is supported through numerical simulations.

q-bio.PE↗

arXiv2015

The scaling limit of a particle system with long-range interaction

We describe the macroscopic behaviour of a particle system with long-range interactions. We describe conditions on the interaction strength in dependency of the distance of the particles, such that the scaling limit of the particle system is a well-posed stochastic PDE.

math.PR↗

arXiv2015

From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step

We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate modelling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population $(K\to \infty)$ size, rare mutations $(u\to 0)$, and small mutational effects $(σ\to 0)$, proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, e.g. by Champagnat and Méléard, we take the three limits simultaneously, i.e. $u=u_K$ and $σ=σ_K$, tend to zero with $K$, subject to conditions that ensure that the time-scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that requires the use of completely different methods. In particular, we cannot use the law of large numbers on the diverging time needed for fixation to approximate the stochastic system with the corresponding deterministic one. To solve this problem we develop a "stochastic Euler scheme" based on coupling arguments that allows to control the time evolution of the stochastic system over time-scales that diverge with $K$.

math.PR↗

arXiv2015

Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime

We prove the convergence of the extremal processes for variable speed branching Brownian motions where the "speed functions", that describe the time-inhomogeneous variance, lie strictly below their concave hull and satisfy a certain weak regularity condition. These limiting objects are universal in the sense that they only depend on the slope of the speed function at $0$ and the final time $t$. The proof is based on previous results for two-speed BBM obtained in a recent paper of ours and uses Gaussian comparison arguments to extend these to the general case.

math.PR↗

arXiv2014

A conditional strong large deviation result and a functional central limit theorem for the rate function

We study the large deviation behaviour of $S_n=\sum_{j=1}^n W_jZ_j$, where $(W_j)_{j \in \mathbb N}$ and $(Z_j)_{j \in \mathbb N}$ are sequences of real-valued, independent and identically distributed random variables satisfying certain moment conditions, independent of each other. More precisely, we prove a conditional strong large deviation result and describe the fluctuations of the random rate function through a functional central limit theorem.

math.PR↗

arXiv2013

The extremal process of two-speed branching Brownian motion

We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is $σ_1$ for $s\leq bt$ and $σ_2$ when $bt\leq s\leq t$. In the case $σ_1>σ_2$, the process is the concatenation of two BBM extremal processes, as expected. In the case $σ_1<σ_2$, a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler.

math.PR↗

arXiv2013

Trait substitution trees on two time scales analysis

In this paper we consider two continuous-mass population models as analogues of logistic branching random walks, one is supported on a finite trait space and the other one is supported on an infinite trait space. For the first model with nearest-neighbor competition and migration, we justify a well-described evolutionary path to the short-term equilibrium on a slow migration time scale. For the second one with an additional evolutionary mechanism-mutation, a jump process-trait substitution tree model is established under a combination of rare mutation and slow migration limits. The transition rule of the tree highly depends on the relabelled trait sequence determined by the fitness landscape. The novelty of our model is that each trait, which may nearly die out on the migration time scale, has a chance to recover and further to be stabilized on the mutation time scale because of a change in the fitness landscape due to a newly entering mutant.

math.PR↗

arXiv2013

Stochastic Modelling of T-Cell-Activation

We investigate a special part of the human immune system, namely the activation of T-Cells, using stochastic tools, especially sharp large deviation results. T-Cells have to distinguish reliably between foreign and self peptides which are both presented to them by antigen presenting cells. Our work is based on a model studied by Zint, Baake, and den Hollander, and originally proposed by van den Berg, Rand, and Burroughs. We are able to dispense with some restrictive distribution assumptions that were used previously, i.e. we establish a higher robustness of the model. A central issue is the analysis of two new perspectives to the scenario (two different quenched systems) in detail. This means that we do not only analyse the total probability of a T-Cell activation (the annealed case) but also consider the probability of an activation of one certain T-Cell type and the probability of a T-Cell activation by a certain antigen presenting cell (the quenched cases). Finally, we see analytically that the probability of T-Cell activation increases with the number of presented foreign peptides in all three cases.

math.PR↗

arXiv2013

Convergence of clock processes in random environments and ageing in the p-spin SK model

We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [(2010), (2011), forthcoming], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [Ann. Probab. 6 (1978) 829-846]. We demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the p-spin SK model. We prove that on a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent results of Ben Arous, Bovier and Cerny [Comm. Math. Phys. 282 (2008) 663-695] that obtained similar convergence results in law, with respect to the random environment.

math.PR↗

arXiv2012

Convergence to extremal processes in random environments and extremal ageing in SK models

This paper extends recent results on aging in mean field spin glasses on short time scales, obtained by Ben Arous and Gun [2] in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in Gayrard [8,9] and naturally complements Bovier and Gayrard [6].

math.PR↗

arXiv2012

An ergodic theorem for the extremal process of branching Brownian motion

In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process.

math.PR↗

arXiv2012

Pointwise estimates and exponential laws in metastable systems via coupling methods

We show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a "meta-stable set." In the example of the Random Field Curie Weiss model, we show that these ideas can also be used to prove asymptotic exponentiallity of normalized metastable escape times.

math.PR↗

arXiv2012

Poissonian statistics in the extremal process of branching Brownian motion

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time $t$. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large $t$ to a random shift of a Poisson point process with exponential density. The last section discusses the Tidal Wave Conjecture by Lalley and Sellke [Ann. Probab. 15 (1987) 1052-1061] on the full limiting extremal process and its relation to the work of Chauvin and Rouault [Math. Nachr. 149 (1990) 41-59] on branching Brownian motion with atypical displacement.

math.PR↗

arXiv2012

Multi-time scales in adaptive dynamics: microscopic interpretation of a trait substitution tree model

We consider a fitness-structured population model with competition and migration between nearest neighbors. Under a combination of large population and rare migration limits we are particularly interested in the asymptotic behavior of the total population partition on supporting trait sites. For the population without mutation on a finite trait space we obtain the equilibrium configuration and characterize the right time scale for fixation. For the model with mutation on an infinite trait space a jump process-trait substitution tree model is established on a rarer mutation time scale against the rare migration constrained in terms of a large population limit. Due to a change of the fitness landscape provoked by a new mutant, some temporarily unfit types can be recovered from time to time. In the end we shed light to illustrate sexual reproduction in a diploid population on the genetic level.

math.PR↗

arXiv2012

An ergodic theorem for the frontier of branching Brownian motion

We prove a conjecture of Lalley and Sellke [Ann. Probab. 15 (1987)] asserting that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a double exponential, or Gumbel, distribution with a random shift. The method of proof is based on the decorrelation of the maximal displacements for appropriate time scales. A crucial input is the localization of the paths of particles close to the maximum that was previously established by the authors [Comm. Pure Appl. Math. 64 (2011)].

math.PR↗