Branching particle systems with interaction, and their scaling limits

我研究的主要主题是研究具有相互作用的分支粒子系统及其缩放限制。我们已经开始朝两个方向努力。第一个项目涉及人口模型在竞争中的波动。在这个模型中，粒子相遇时以c的速率分支、扩散和聚合。由于增长是线性的，而竞争是二次的，所以该模型收敛到均衡状态。当c趋近于0时，平衡种群以1/c的速度增长。在这种情况下，我们希望了解这个过程在其平衡附近的波动。当一个粒子分支时，它会产生随机数量的后代。如果该复制律存在第二矩，则适当重标的波动过程收敛为具有耗散和加性高斯白噪声的随机热方程。分支为二元时，期望水动力极限由FKPP方程给出，且宏观尺度上的波动满足随机索方程。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, Walsh 1981, Fox 1987）等不同领域具有潜在的兴趣。这个过程也是随机FKPP方程的对偶（Doering et al. 2003）。另一方面，如果第二力矩不存在，那么我们预计会看到列维噪声驱动波动的动力学。我们进一步的目的是了解当平均值为无穷大时，可能在非爆炸条件和研究Yaglom型极限之后会发生什么。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的，分支事件通过移除粒子来抵消。例子包括布朗蜜蜂模型（J. Berestycki et al. 2020, Addario-Berry et al. 2020），弗莱明-维奥粒子系统（Fleming et al. 1979）和N-BBM （N. Berestycki et al. 2013, Groisman et al. 2019, De Masi et al. 2017）。我们感兴趣的是当种群大小趋于无穷大时粒子密度的宏观行为（流体动力极限、波动）、系统的长期行为以及模型参数（空间维度、子代分布、选择规则）所起的作用。2017年，De Masi等人在一个空间维度上通过杀死最左边的粒子计算了N-BBM的水动力极限。极限点用自由边界问题（FBP）来描述，J. Berestycki等人在2018年证明了其全局存在性。我们希望研究De Masi等人的猜想，即对于由最左边的粒子重新进入中心的N- bbm存在强选择原理，即重新进入中心的粒子系统的不变测度的大N极限是一个连续的概率测度，其密度由流体动力极限的最小行波剖面给出。接下来我们介绍伯努利布朗蜜蜂，在分支事件中，离原点最近的粒子以p的概率被移除，离原点最远的粒子以1-p的概率被移除。如果p<1/2，那么我们期望有流体动力极限的平稳解，以及对N只蜜蜂的过程的不变测度。我们的主要目标是证明选择原则成立，根据空间维度和p具有不同的宏观行为。正如Bramson等人1986所强调的那样，这些问题可能有助于阐明模式形成和选择的普遍规则。此外，它们与fbp的研究有关，它们提供了粒子系统方面的解决方案的表示。该项目属于EPSRC的以下研究领域：数学分析、统计和应用概率。

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.