Probabilistic methods in KPZ universality and stochastic optimisation

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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.