超过程是分枝马氏过程的高密度极限过程，在解决诸多实际问题中有着广泛的应用和需求，因此，超过程成为近年来概率论及偏微分方程等方向的研究热点之一，受到了广泛的关注和深入研究。我们计划研究具有一般底过程及分枝机制（包括非局部分枝机制）的超过程的强大数律、中心极限定理、中偏差与大偏差结果等。我们还将研究随机环境中的超过程的极限理论，并用概率方法研究与超过程相关的非线性偏微分方程的解结构。研究具有非局部分枝机制的超过程首先需要对底过程的无穷小生成元算子在非局部算子扰动下的性质进行研究。我们将把对非局部扰动的研究成果应用到对具有一般分枝机制的超过程的研究上。

Superprocesses are high density limits of branching Markov processes. Superprocesses arise naturally in many applications and thus have been intensively studied in recent years not only by probabilists, but also by researcher in other fields such as partial differential equations. We propose to study limit properties, including strong law of large numbers, central limit theorems, moderate and large deviations, of superprocesses with general spatial motions and general branching mechanism (including non-local branching mechanism). We will also study limit theorems for superprocesses in random environments, and structure of solutions of nonlinear differential equations related to superprocesses. For the study of superprocesses with non-local branching mechanism, one needs to study the infinitesimal generator of the underlying spatial motion under non-local perturbation. We will apply the results of non-local perturnation to the study of superprocesses with general branching mechanism.