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

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).