近年来，超过程的谱系结构的研究受到很多学者的重视。本项目将研究具有一般分枝机制的超过程的Williams分解，刻画超过程在灭绝时刻为h时的谱系结构。Delmas和Hénard (2013)解决了分枝机制为二分枝的超过程的Williams分解，我们将对其结论进行推广。同时作为Williams分解的一个重要应用，我们将研究具有一般分枝机制的超过程在趋近灭绝时刻时的极限行为。本项目还将研究分枝过程和超过程的大偏差和中偏差定理，主要包括以下三方面内容：1) 研究带移民的下临界多物种连续时间分枝过程的大偏差定理和中偏差定理; 2) 解决带移民的具有一般分枝机制的下临界超过程的占位时的大偏差定理和中偏差定理，推广Hong和Li(2005)中的结论; 3) 研究上临界分枝马氏过程和超过程的大偏差和中偏差定理。

The genealogy of superprocesses has been studied by many people in recent years. Williams' decomposition characterizes the genealogy structure of superprocesses conditioned on the extinction time being h. Delmas and Henard (2013) established a Williams' decomposition for superprocesses with binary branching mechanisms. In this project, we will study Williams' decomposition for superprocesses with general spatial dependent branching mechanisms. As an important application of Williams' decomposition for superprocesses, we will study the limiting behavior of superprocesses, with general spatial dependent branching mechanisms, near extinction. We will also investigate moderate and large deviations for branching processes and superprocesses, including the following three problems: 1) study moderate and large deviations for subcritical multitype continuous time branching processes with immigration; 2) investigate moderate and large deviations for occupation times of subcritial superprocesses with immigration and generalize the results of Hong and Li (2005) which dealt with superprocesses with spatial independent binary branching mechanisms; 3) study moderate and large deviations for supercritical branching Markov processes and superprocesses.