Random walks and branching processes in random environments under Spitzer's condition

随机游动和分支过程构成了随机过程理论的两个基本基石。第一个模型是离散时间内随机运动粒子的演化，第二个模型是离散时间内无性繁殖种群的演化，这两类过程之间有许多密切的关系。特别是通过路径变换和/或极限定理。这使得它们非常有吸引力的数学对象，其中发生在它们之间的交互现象往往为更深入和一般的数学研究对象的发展提供了平台，这些对象本身导致了许多数学应用。目前的提案旨在资助分支过程领域的主要专家之一V. Vatteri教授进行为期两个月的访问，其主要目的有两个。1.通过对随机环境中的分支过程和随机环境中的分支过程的研究，建立了随机环境中随机游动的新结果，即随机环境中的随机游动.为配合新成立的麦克斯韦数学科学研究所，Vattera教授将主讲经典分支过程的现代观点（包括随机环境中的分支过程和分支过程在研究进化系统和种群生物学中的应用）。

Random walks and branching processes form two fundamental cornerstones of the theory ofstochastic processes. The first models the evolution of a randomly moving particle in discrete time and the second (in its most basic form) modelsthe evolution in discrete time of an asexually reproducing population.There are many intimate relationships between these two classes of processes. In particular via path transformations and/or limit theorems.This makes them very attractive mathematical objects in which the interactive phenomena that occur between them has often provided a platform for the development of much deeper and generic mathematical objects of study which themselves lead on to a number of mathematical applications. The current proposal aims at funding a two month visit of one of the leading experts in the field of branching processes, Prof. V. Vatutin, with a view to two main objectives.1. Establishing new results concerning random walks whose probabilistic transitions are governed by a mathematical mechanism whichitself is randomised (the so called random walk in random environment) by drawing conclusions frombranching processes in random environments and vice versa.2. In light of the creation of the new Maxwell Institute for Mathematical Sciences, Prof. Vatutin will give a special lecture course on modern aspects of classical branching processes (including branching processes in random environment and application of branching processes to study queueing systems and some problems inpopulation biology).