Stochastic interacting systems: connections, fluctuations and applications

概率论中的许多经典结果都考虑了独立的(或弱相依的)且通常是同分布的随机变量。在这些情况下，关于样本平均的收敛(大数定律)及其波动(中心极限定理)的最基本问题是很好理解的。当失去独立性时，情况就会变得非常不同。人们发明了几个模型来帮助我们理解不同现象中的观察结果，如凝聚态和统计物理、分子、细胞水平和人口生物学、地理学、社会学和工程学。这里考虑的过程的一个共同特征是失去独立性(或弱依赖)。物理系统中的原子传输过程相互作用，分子在细小的细胞通道中进行传输时也是如此，或者在人体狭窄血管中的血细胞也是如此。工程技术人员看到了我们街道上正在发生的交通流量参与者之间的类似互动。生物的趋化性和其他自主运动取决于移动细胞的局部环境，使其行为与看到相同环境的任何较早的路径段相关。人口中的感染或树木中的森林火灾以随机生长的表面的形式发展，其中增长的速度取决于当地社区中完全相同的表面的形状。雪崩被模拟为雪块的自增强相互作用的随机运动，而河床一段的变化强烈地影响其他部分的变化。投票意见和谣言传播显然是一个高度交叉关联的社会学过程，具有大量的空间互动。数据传输系统使用的队列通过路由算法以复杂的方式进行交互。在上述所有例子中，任何试图建立随机观测模型的尝试都会很快导致随机变量序列的随机相依。这些过程中的许多时间和/或空间相关性使得经典方法不可能应用。在某些情况下，可以发明新的想法来证明大数定律和中心极限定理的行为。在其他一些情况下，中心极限定理和作为普适极限的正态分布的标度将不再有效。新的、仍然非常普遍的标度级和极限分布出现了，这是相互作用过程的一般类别的特征，但与通常的独立图景本质上不同。最近几年取得了一些开创性成果的一个例子是所谓的Kardar-Parisi-Zhang方程，它具有特征时间^{1/3}标度和Tracy-Widom极限分布。因此，在这些领域的数学研究需要本质上的新想法，这些想法经常与数学(泛函分析、代数、组合数学、动力学系统等)和物理的各个领域强烈地相互作用。我们的研究针对上述一些模型中的结构、平稳行为、涨落和尺度极限等基本问题。更具体地说，我们调查：-在固定和动态变化的随机环境中的随机游动，在这种环境中，原本简单的随机运动根据其位置改变其行为；-相互作用的粒子系统，其中许多本来简单的运动相互作用。我们的每个研究问题都涉及与经典的成熟场景相去甚远的情况。为了得出结果，我们使用了来自其他数学领域的原始概率思想和工具。我们可以在这些系统中展示的行为是新的，并极大地提高了我们对使用这些模型的其他科学的一般理解。我们的工作也代表着对数学的宝贵贡献，因为我们与其他领域的互动，以及一个人在缺乏传统工具的情况下必然发明的各种想法。

Many classical results in probability theory consider independent (or weakly dependent) and often identically distributed random variables. The most fundamental questions regarding convergence of the sample average (Law of Large Numbers) and the fluctuations thereof (Central Limit Theorem) are well understood in these cases. The picture becomes very different when independence is lost.Several models have been invented to help our understanding of observations in diverse phenomena such as condensed matter and statistical physics, molecular, cell-level and population biology, geography, sociology and also engineering. A common feature of the processes considered here is the loss of independence (or weak dependence). Atomic transport processes in physical systems interact with each other, as well as molecules do when progressing in small cellular channels, or blood cells in narrow vessels of the body. Engineering sees similar interactions between participants of traffic flow that is happening on our streets. Chemotaxis and other autonomous motion of biology depend on the local environment of the moving cell, making its behaviour correlated to any earlier segment of the path that saw the same environment. Infections in a population or forest fires among trees advance as a randomly growing surface where the speed of growth depends on the shape of the very same surface in a local neighbourhood. Avalanches are modeled as self-reinforcing interacting random motion of blocks of snow, while changes in one stretch of a riverbed strongly influence those in other bits. Voting opinions and rumour spreading is obviously a very cross-correlated process of sociology with lots of spatial interactions. Data-transmission systems use queues that interact in complicated ways via routing algorithms. In all of the above examples any attempts to build a stochastic model of observations quickly lead to stochastically dependent sequences of random variables.The temporal and/or spatial dependence in many of these processes makes it impossible to apply the classical methods. In some instances new ideas can be invented to prove the Law of Large Numbers and Central Limit Theorem behaviour. In some other cases the scaling of the Central Limit Theorem and the Normal distribution being the universal limit will simply not be valid anymore. New, still very universal scaling orders and limit distributions emerge, characteristic to general classes of interacting processes but essentially different from the usual independent picture. An example where several groundbreaking results have been achieved in recent years is the so-called Kardar-Parisi-Zhang equation with its characteristic time^{1/3} scaling and Tracy-Widom limit distributions.Mathematical research in these areas thus require essentially new ideas that often strongly interact with various fields of mathematics (functional analysis, algebra, combinatorics, dynamical systems among others) and physics. Our research aims at fundamental questions of constructions, stationary behaviour, fluctuations and scaling limits in some of the above models. More specifically, we investigate:- random walks both in fixed and dynamically changing random environments, where an otherwise simple random motion changes its behaviour depending on its position;- interacting particle systems, where many, otherwise simple motions interact with each other.Each of our research questions concerns cases that are far from the classical well-established scenarios. To come up with results we apply original probabilistic ideas and tools from other fields of mathematics. The behaviour we can demonstrate in these systems is new, and greatly improves our general understanding in other sciences which use these models. Our work also represents valuable contributions to mathematics because of interactions with other areas and because of the variety of ideas that one necessarily invents in the lack of traditional tools.