Branching particle systems with interaction, and their scaling limits

我的研究主题是研究具有相互作用的分支粒子系统及其标度极限。我们已经开始了两个方向的工作。第一个项目涉及的竞争人口模型的波动。在该模型中，粒子相遇时以速率c进行分支、扩散和合并。由于增长是线性的，而竞争是二次的，该模型收敛到一个平衡状态。当c为0时，均衡种群的增长率为1/c。在这种情况下，我们想了解这个过程在其平衡点附近的波动。当粒子分支时，它会产生随机数量的后代。如果二阶矩存在的再生法，然后适当重新标度的波动过程收敛到一个随机热方程的耗散和加性高斯白色噪声。当分支是二进制的，它是预期的流体动力学极限是由FKPP方程，并在宏观尺度上的波动满足随机电缆方程。这在随机量化（Parisi et al. 1981）、治疗期间肿瘤生长的量化（Swanson et al. 2003）和计算神经科学（Tuckwell 1989，沃尔什1981和Fox 1987）等不同领域都具有潜在的意义。这个过程也是随机FKPP方程的对偶（Doering et al. 2003）。另一方面，如果二阶矩不存在，那么我们期望看到Levy噪声驱动波动的动力学。我们进一步的目标是了解当平均值无限时会发生什么，可能是在非爆炸条件下并研究Yaglom型极限之后。第二个研究方向涉及具有扩散、分支和选择的粒子系统。在这些模型中，种群的大小是固定的或由一个参数控制，分支事件被粒子的移除所抵消。例子包括布朗蜜蜂模型（J. Berestycki et al. 2020，Addario-Berry et al. 2020）、Fleming-Viot粒子系统（Fleming et al. 1979）和N-BBM（N. Berestycki et al. 2013，Groisman et al. 2019，De马西et al. 2017）。我们感兴趣的是当粒子数趋于无穷大时粒子密度的宏观行为（水动力极限、波动）、系统的长时间行为以及模型参数所起的作用（空间维度、后代分布、选择规则）。2017年，De马西等人计算了N-BBM在一维空间中杀死最左边粒子的流体动力学极限。极限点由自由边界问题（FBP）描述，其全局存在性由J. Berestycki等人在2018年证明。我们希望研究De马西等人的猜想，即强选择原理适用于由最左粒子重心的N-BBM，在这个意义上，重心粒子系统的不变测度的大N极限是一个连续的概率测度，其密度由流体动力学极限的最小行波的轮廓给出。接下来我们介绍伯努利布朗蜜蜂，其中在分支事件中，最接近原点的粒子以概率p被移除，而距离原点最远的粒子以概率1-p被移除。如果p<1/2，那么我们期望有流体动力学极限的稳定解，以及N只蜜蜂上过程的不变测度。我们的主要目标是证明，选择原则，不同的宏观行为取决于空间维度和p.在Bramson等人1986年强调，这些问题可能有助于揭示模式形成和选择的普遍规则。此外，它们与FBPs的研究有关，它们提供了粒子系统的解决方案。该项目属于以下EPSRC研究领域：数学分析，统计和应用概率福尔斯。

The main theme of my research is the study of branching particle systems with interaction, and their scaling limits. We have started working in two directions.A first project concerns the fluctuations of a model of population with competition. In this model, particles branch, diffuse, and coalesce at a rate c when they meet. As the growth is linear while the competition is quadratic, this model converges to an equilibrium state. When c is sent to 0, the equilibrium population grows like 1/c. In that setting, we would like to understand the fluctuations of the process around its equilibrium. When a particle branches, it produces a random number of offsprings. If a second moment exists for this reproduction law, then the appropriately rescaled fluctuations process converges to a stochastic heat equation with dissipation and additive Gaussian white noise. When branching is binary, it is expected that the hydrodynamic limit is given by the FKPP equation, and that the fluctuations at the macroscopic scale satisfy the stochastic cable equation. This is of potential interest in areas as diverse as stochastic quantization (Parisi et al. 1981), the quantification of tumor growth during treatment (Swanson et al. 2003), and computational neuroscience (Tuckwell 1989, Walsh 1981, and Fox 1987). This process is also the dual to the stochastic FKPP equation (Doering et al. 2003). On the other hand, if a second moment does not exist, then we expect to see a Levy noise driving the dynamics of the fluctuations. We further aim to understand what happens when the mean is infinite, possibly after conditioning on non-explosion and studying Yaglom type limits. A second direction of research concerns particle systems with diffusion, branching and selection. In those models, the size of the population is fixed or controlled by a parameter and branching events are offset by the removal of particles. Examples include the Brownian bees model (J. Berestycki et al. 2020, Addario-Berry et al. 2020), the Fleming-Viot particle systems (Fleming et al. 1979), and the N-BBM (N. Berestycki et al. 2013, Groisman et al. 2019, De Masi et al. 2017). We are interested in the macroscopic behavior of the density of particles when the population size goes to infinity (hydrodynamic limit, fluctuations), the long-time behavior of the system, and the role played by the parameters of the model (spatial dimension, offspring distribution, selection rule).In 2017, De Masi et al. computed the hydrodynamic limit of the N-BBM with killing of the leftmost particle in one spatial dimension. The limit point is described by a free boundary problem (FBP), whose global existence was shown by J. Berestycki et al. in 2018. We wish to work on the conjecture of De Masi et al. that a strong selection principle holds for this N-BBM recentered by the leftmost particle, in the sense that the large N limit of the invariant measure of the recentered particle system is a continuous probability measure with density given by the profile of the minimal travelling wave of the hydrodynamic limit. We next introduce the Bernoulli Brownian bees, where at a branching event, the particle closest to the origin is removed with probability p, and the particle furthest from the origin is removed with probability 1-p. If p<1/2, then we expect to have stationary solutions of the hydrodynamic limit, and an invariant measure for the process on N bees. Our main goal is to prove that a selection principle holds, with different macroscopic behaviour depending on the spatial dimension and on p. As highlighted in Bramson et al. 1986, these problems may help shed light on universal rules in pattern formation and selection. In addition, they are relevant to the study of FBPs where they provide representations of solutions in terms of particle systems. This project falls within the following EPSRC research areas: mathematical analysis, and statistics and applied probability.