Stochastic processes with spatial constraints

该建议侧重于无限随机系统的演变是空间约束。 它涉及时空非齐次接触过程和分支随机游动的研究。 它还涉及准稳态分布的研究，这是一个重要的概念，涉及到一个过程的平衡性质的条件下，留在一个子状态空间。在较早的工作中，我们建立了准平稳分布和Fleming-Viot粒子系统之间的关系，这承诺了一个新的，建设性的和有用的方法准平稳性。拟议的关于这一主题的研究将在这项工作的基础上进行，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，为了使它们更容易访问，人们将随机分量分配给这些相互作用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。它们也被发现是流行病学中非常有用的模型。 拟议的项目是有关无限随机系统的演变是空间约束。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人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？除了其数学意义外，所提出的研究在地理学和进化理论中具有非常重要的应用，希望它将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.