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

奇异类测值随机过程及相关非线性系统的基础研究。

• 批准号: 14540101
• 负责人: DOKU Isamu
• 金额: $ 2.24万
• 依托单位: SAITAMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540101/
• 关键词: Measure-valued stochastic processes; Branching processes; Random measures; Interactive particle systems; Immigration superprocesses; Scaling; Limit theorem; Convergence problem; 測定値確率過程; ランダム測定; 超ブラウン運動; 確率偏微分方程式; 法則の絶対連続性; 測度値過程; 分枝マルコフ過程 /

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.

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支率，存在唯一的测度值分支马氏过程，其中包括具有双曲分支率的Fleischmann-Muller超过程.我们还展示了它们路径的规律性。然后给出了这类超过程具有有限时间绝灭性的一个充分条件。进一步，我们将这些结果推广到一类具有扩散型空间运动的广义超过程，并将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超过程，证明了在测度值连续路径空间上，重标移民超过程在分布意义下收敛于SCSM的极限定理.目前，我们正在计划一个新的研究项目，我们的目标是得到一个新的极限定理，其中一个比SCSM更广义的超过程作为一个重标极限出现，并且我们证明了与这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，我们应用随机分析证明了解的规律的绝对连续性。

项目成果

Mutually interactive type of measure-valued processes and approximation scheme.

相互交互类型的测值过程和近似方案。

• 发表时间: 2004
• 期刊: J.Saitama Univ.Fac.Educ.(Math.Nat.Sci.) vol.53,no.1
• 作者: Doku;I.;Isamu DOKU;Isamu DOKU;Isamu DOKU;Isamu Doku
• 通讯作者: Isamu Doku

Some nonlinear elliptic boundary value problems and killing property for singular superprocesses.

奇异超级过程的一些非线性椭圆边值问题和杀伤性。

• 发表时间: 2003
• 期刊: J.Saitama Univ.Fac.Educ.(Math.Nat.Sci.) vol.52,no.2
• 作者: Doku;I.;Isamu DOKU;Isamu DOKU;Isamu Doku;Isamu Doku;Isamu Doku
• 通讯作者: Isamu Doku

Isamu DOKU: "Historical Approach to the support problem for measure-valued processes associated with nonlinear stochastic equations"J. Saitama Univ. Math. Nat. Sci.. 51・1. 1-12 (2002)

Isamu DOKU：“与非线性随机方程相关的测值过程的支持问题的历史方法”J. Nat. 51・1。

A system of stochastic partial differential equations driven by a coloured noise and the martingale problem for symbiotic branching superprocesses.

由有色噪声和共生分支超级过程的鞅问题驱动的随机偏微分方程组。

• 发表时间: 2006
• 期刊: J.Saitama Univ.Fac.Educ.(Math.Nat.Sci.) vol.55,no.1
• 作者: Isamu Doku

On the construction of superdiffusions with singular branching rate via regularized approximation.

通过正则化近似构建具有奇异分支率的超扩散。

• 发表时间: 2003
• 期刊: J.Saitama Univ.Fac.Educ.(Math.Nat.Sci.) vol.52,no.2
• 作者: Doku;I.;Isamu DOKU;Isamu DOKU;Isamu Doku;Isamu Doku
• 通讯作者: Isamu Doku