Probabilistic methods in KPZ universality and stochastic optimisation

建议的研究属于概率论领域，即研究不确定条件下的现象。全球目标是更好地了解随机事件，特别是它们的分布和长期行为。在这项建议中，我们关注这一领域的理论和应用两个方面。一个共同的主题是寻求优化受随机约束的数量，以及研究不确定性在这种背景下的影响和好处的必要性。 为了给出一种所研究的现象的具体例子，考虑最后一段渗流，它可以被视为液体在多孔环境中运动的模型：步行者穿过网格，在网格的每个点上放置随机数的奖励。步行者的目标是选择一条路径，以最大化她在旅途中获得的总回报。如果奖励是随机放置的，服从给定的概率分布，那么最优路径和最大可能的奖励本身就是随机变量。研究这些随机变量，特别是在起点和终点之间的距离变得更大的情况下，是这一提议的一个关键目标。一个令人惊讶的发现是，这种长期行为并不取决于所选择的具体奖励分配。这是一种被称为普遍性的现象的一个例子，在液晶生长、细菌菌落生长和火灾传播中都观察到了这种现象。一系列令人惊讶的数学技术被应用到它的研究中，包括组合学、代数、分析和排队论。 一个相关的问题涉及寻找一个数量的最小值，例如，在随机约束下，例如通过网络运输货物的成本，例如对要运输的货物的需求。这个问题通常用数值方法(借助计算机程序)来解决，而不确定的参数则用有限数量的情景来表示。然而，为了准确地表示潜在的不确定性，必须对大量的情景进行抽样。除了计算成本高昂之外，这种高度的复杂性还会使决策者很难理解模型是如何选择特定的决策的。在这份提案中，我将在复杂性和准确性之间做出新的权衡。一个关键因素是机会成本，即在预测错误情景后做出决定的成本。 同样，大型基础设施项目的管理者往往需要根据不确定的未来现金流做出最优投资决策。为了帮助他们做出更好的决策，事实证明，将投资的可能性建模为一种财务选择是有帮助的。虽然与金融期权理论有相似之处，但也有显著的差异，因为这两种理论遵循的假设截然不同。在拟议的研究中，我将通过创新的概率方法来更好地理解这一模型。

The proposed research is situated in the field of probability theory, that is the study of phenomena under uncertainty. The global objective is to better understand random occurrences, particularly their distribution and their long-term behaviour. In this proposal, we are concerned both with theoretical and applied aspects of this field. A common theme is the quest to optimise a quantity subject to random constraints, and the need to study the influence and benefits of uncertainty in this setting. To give a specific example of the kind of phenomena studied, consider last-passage percolation, which can be considered as a model for the movement of liquids through a porous environment: a walker crosses a grid, with random numbers of rewards placed on each point of the grid. The aim of the walker is to choose a path so as to maximise the total reward she picks up along the way. If the rewards are placed randomly, subject to a given probability distribution, the optimal path and the largest possible rewards are themselves random variables. Studying these random variables, particularly as the distance between start and end-point becomes larger, is a key objective of this proposal. A surprising finding is that this long-term behaviour doesn't depend on the specific reward distribution chosen. This is an example of a phenomenon called universality and has been observed in liquid crystal growth, bacterial colony growth and fire propagation. A surprising array of mathematical techniques have been applied to its study, including combinatorics, algebra, analysis and queueing theory. A related problem concerns finding the minimum of a quantity, for example the cost of transporting goods across a network, subject to random constraints, such as the demand for goods to be transported. This problem is often solved numerically (with the help of computer programs) and the uncertain parameters are represented by a finite number of scenarios. However, in order to accurately represent the underlying uncertainty, a large number of scenarios must be sampled. Apart from being computationally expensive, this high complexity can also make it difficult for decision makers to understand how a particular decision was chosen by the model. In this proposal, I will produce new trade-offs between complexity and accuracy. A crucial ingredient are opportunity cost, that is the cost of making a decision after predicting the wrong scenario. Similarly, managers of large infrastructure projects often need to make optimal investment decisions based on uncertain future cash flows. In order to help them make better decisions, it has proved to be helpful to model the possibility to invest as a financial option. While there are similarities to the theory of financial options, there are significant differences, because the two follow very different assumptions. In the proposed research, I will contribute to a better understanding of this model with innovative probabilistic approaches.