Basic Research for a Singular Class of Measure-Valued Stochastic Processes and Associated Nonlinear Systems.

我们的主要兴趣是一类奇异超过程。首先，我们证明了对于更广泛的一类奇异分枝率，存在唯一的测值分枝马尔可夫过程，其中包括一个具有双曲分枝率的Fleischmann-Muller超过程。我们还展示了它们的路径的规律性。然后，我们得到了这类超过程具有有限时间灭绝性的一个充分条件。此外，我们将这些结果推广到一类具有类扩散空间运动的广义超过程，并尝试将Dawson-Li-zhon收敛结果推广到SCSM。实际上，我们考虑了一类更复杂的带移民的超过程，证明了重定标移民超过程在测值连续路空间上的分布意义下收敛于SCSM的一个极限定理。现在我们正在计划一个新的研究项目，目标是推导出一种新型的极限定理，其中一个比SCSM更广义的超过程作为重标度极限出现。此外，我们证明了与这样一类奇异超过程相关的各种随机方程弱解的存在唯一性定理。对于一个具体的例子，我们用随机分析的方法证明了解律的绝对连续性。

Our main interest is a class of singular superprocesses. First of all we proved that there exists a unique measure-valued branching Markov process for a wider class of singular branching rates, which includes a Fleischmann-Muller superprocess with hyperbolic branching rate. We also showed the regularity of their paths. Then we derived a sufficient condition for such a class of superprocesses to possess the finite time extinction property. Moreover, we extended those results to a class of generalized superprocesses with diffusion-like spatial motion.Next we tried to obtain a generalization of Dawson-Li-Zhon convergence result to SCSM. Actually we considered a more complicated class of superprocesses with immigration, and proved a limit theorem that rescaled immigration superprocesses converges to a SCSM in the sense of distribution on the space of measure-valued continuous paths. Now we are planning a new research project that we aim at deriving a new type of limit theorem, where a superprocess more generalized than SCSM arises as a rescaled limit.Furthermore, we proved an existence and uniqueness theorem for weak solution to various kinds of stochastic equations associated with such a class of singular superprocesses. For a specific example, we showed absolute continuity of the law of solution by applying stochastic analysis.