本学位论文在经典超布朗运动及带吸收分枝布朗运动的基础上，系统地研究了带吸 收超布朗运动的长时间渐近行为．论文主要分为三个部分．

在第一部分中，我们研究带吸收超布朗运动的生存概率．在 Kyprianou 及其合作者 2012 年的研究工作中 ，他们通过骨架分解分析了一类分枝上临界的带吸收超布朗运动的 行为．在本论文中我们将在下临界漂移条件下研究带吸收超布朗运动的生存概率，并通过 分析一类具有奇异边界条件的非线性偏微分方程的解给出生存概率的衰减速度．进而我 们将在过程存活的条件下研究其条件极限分布，并且证明相应的 Yaglom 极限定理．

在第二部分中，我们研究带吸收超布朗运动的大偏差概率．这个问题研究的是支撑集 的上确界与其极限值有一定偏差的概率的衰减速度．我们将通过一类具有奇异边界条件 的偏微分方程的解刻画大偏差概率，并进一步利用布朗桥和测度变换等给出偏差概率的 长时间渐近行为作为一个副产品，我们还将证明相应的 Yaglom 条件极限定理.

在第三部分中，我们研究超布朗运动的吸收质量．这个问题考虑的是与吸收壁位置 相关的退出测度上的质量的尾分布．我们通过拉普拉斯变换和微分方程刻画吸收质量过 程的累积半群，并进一步通过奇异性分析的方法分析累积半群在奇点附近的渐近性质．最 后我们根据陶伯定理给出吸收质量尾分布的衰减速度．

Based on super-Brownian motion and branching Brownian motion with absorption, this thesis systematically studies the long-time asymptotic behavior of super-Brownian motion with absorption. This thesis consists of three parts.

In the first part, we study the survival probability for super-Brownian motion with absorption. Kyprianou et al. (2012) analyzed the behavior of supercritical super-Brownian motion with absorption through the backbone decomposition. In this thesis, we investigate the survival probability in subcritical drift case and give the decay rate of the survival probability by analyzing the solution of a class of nonlinear partial differential equation with singular boundary conditions. Furthermore, we study the conditional limit distribution of the process under the condition of survival and prove the corresponding Yaglom limit theorem.

In the second part , we study the large deviation probability for super-Brownian motion with absorption. This problem studies the decay rate of the probability that the supremum of the support deviates from its limit value. We characterize the large deviation probability through the solution of a class of partial differential equations with singular boundary conditions and further use Brownian bridges and measure transformations to provide the long-time asymptotic behavior of the deviation probability. As a by-product, a related Yaglom-type conditional limit theorem is obtained.

In the final part, we study the absorbed mass for super-Brownian motion. This problem studies the tail distribution of mass on the exit measure associated with the absorption. We characterize the cumulant semigroup of the absorbed mass process by its Laplace transform and differential equation. Furthermore, we analyze the asymptotic properties of the cumulant semigroup near the singularity. Finally, applying Tauberian theorem we obtain the decay rate of the tail distribution of the absorbed mass.