Stochastic processes with spatial constraints

该方案关注的是演化受空间约束的无限随机系统。它涉及时空非均匀接触过程和分支随机漫步的研究。它还涉及到准平稳分布的研究，准平稳分布是一个基本概念，与一个过程的平衡性质有关，这个过程被限制在一个子状态空间内。在早期的工作中，我们建立了准平稳分布与弗莱明-维奥粒子系统之间的关系，这为准平稳提供了一种新的、建设性的和有用的方法。关于该主题的拟议研究将建立在这项工作的基础上并大大扩展这项工作。许多自然现象和社会现象是大量成分（粒子、人类、病毒、植物等）相互作用的结果。由于这些系统的巨大复杂性，为了使它们更容易接近，人们给这些相互作用分配随机成分，并研究相应的随机模型。相互作用随机系统作为各种物理系统的合适模型，在过去的几十年里得到了广泛的研究。它们也被认为是流行病学中非常有用的模型。所提出的项目涉及无限随机系统，其演化是空间约束的。一些需要解决的数学问题可以翻译为：如果感染只通过一定数量的个体和他们之间的接触传播，会发生什么？栖息地的减少如何影响植物物种的分布？除了具有数学意义外，本研究在生物地理学和进化论方面也具有重要的应用价值，并有望产生富有成效的跨学科合作研究。

The proposal focuses on infinite stochastic systems whose evolution is spatially constrained. It involves research on space-time inhomogeneous contact processes and branching random walks. It also involves a study of quasi-stationary distribution which is an essential concept related to equilibrium properties of a process conditioned to stay inside a sub-state-space. In an earlier work, we established a relation between quasi-stationary distributions and a Fleming-Viot particle system, which promises a novel, constructive and useful approach to quasi-stationarity. The proposed research on the subject will build on and significantly extend this work. Many natural and social phenomena arise as a result of interaction of large number of components (particles, humans, viruses, plants,...). Due to immense complexity of these systems, in order to make them more accessible, one assigns random components to these interactions and studies corresponding stochastic models. As suitable models for various physical systems, interacting stochastic systems have been studied extensively over the last decades. They were found also to be very useful models in epidemiology. The proposed project is related to infinite stochastic systems whose evolution is spatially constrained. Some mathematical questions to be addressed can be translated as:What happens with an infection if it spreads using only a certain number of individuals and contacts among them? How does reduction of a habitat affect distributions of a plant species? Beside its mathematical significance, the proposed research has very important applications in biogeography and evolution theory and hopefully it will generate fruitful collaborative interdisciplinary research.