Research on several problems of stochastic processes concerning the Ginzburg-Landau continuum model

Ginzburg-Landau连续介质模型随机过程几个问题的研究

• 批准号: 11640111
• 负责人: SUGIURA Makoto
• 金额: $ 0.96万
• 依托单位: University of the Ryukyus
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640111/
• 关键词: continuum model; Gibbs measure; mixing condition; log-Sobolev inequality; stochastic calculus; Dirichlet form; Ginzburg-Landau連続場模型; storage process; Dobrushin-Shlosman型混合条件

Through the present research project, we obtain the following results.0. In Osada and Spohn's paper, they discuss the existence and uniqueness problems of the Gibbs measure concerning the Ginzburg-Landau continuum model. Their assumptions for the self-potential and interaction-potential are very mild. The sufficient condition of the uniqueness is the same as that for the lattice case derived by Papangelou. They also give an example in which the phase translation occurs.Recently, Hariya extended this results to the case with reflection condition.1. Hariya and Osada construct a stochastic process corresponding to the Gibbs measure defined above. They consider the case when the time evolution is corresponding to the Malliavin stochastic calculus. Since they use the Dirichlet form, the assumption for the potential is very mild.2. Sugiura investigates the mixing property for the Gibbs state constructed above in O.He derives the Dobrushin-Shlosman type mixing property. Since the model is continuum, the decay of the correlation of near spins is also important. His result includes this property which is the best possible one. He also considers the stochastic process perturbed by cylindrical Brownian motion and obtains a partial results about the log-Sobolev inequality for this model. The paper of the corresponding results is in preparation. He also reduces this problem to some estimates of covariance. Doing this is the next research project for us.

通过本课题的研究，取得了以下成果.在Osada和Spohn的论文中，他们讨论了Ginzburg-Landau连续模型的Gibbs测度的存在唯一性问题。他们对自势和相互作用势的假设是很温和的。唯一性的充分条件与Papangelou导出的格情形相同。最近，Hariya将这一结果推广到了具有反射条件的情形. Hariya和Osada构造了一个对应于上面定义的Gibbs测度的随机过程。他们认为的情况下，时间演化对应的Malliavin随机微积分。由于他们使用的是狄利克雷形式，对势的假设是非常温和的. Sugiura研究了O.上面构造的Gibbs态的混合性质，他推导出了Dobrushin-Shlosman型混合性质。由于该模型是连续的，近自旋相关的衰减也很重要。他的结果包括这个性质，这是最好的可能。他还考虑了受圆柱布朗运动扰动的随机过程，得到了关于该模型的log-Sobolev不等式的部分结果。相应结果的论文正在编写中。他还减少了这个问题的一些估计的协方差。这是我们的下一个研究项目。

项目成果

Taizo Chiyonobu: "A limit formula for a class of Gibbs measures with long range interactions"J.Math.Sci.Univ.Tokyo. 7, no.3. 463-486 (2000)

Taizo Chiyonobu：“具有长程相互作用的一类吉布斯测度的极限公式”J.Math.Sci.Univ.Tokyo。

Kazunori Kodaka: "FS-property for C^*-algebras"Proc.Amer.Math.Soc..

Kazunori Kodaka：“C^*-代数的 FS 性质”Proc.Amer.Math.Soc..

Taizo Chiyonobu: "A limit formula for a class of Gibbs measures with long range interactions"J.Math.Sci.Univ.Tokyo. 7-3. 463-486 (2000)

Taizo Chiyonobu：“具有长程相互作用的一类吉布斯测度的极限公式”J.Math.Sci.Univ.Tokyo。

Kazunori Kodaka: "The positive cones of Ko-groups of crossed products associated with Furstenberg transformations of the 2-torus"Proc.Edinburgh Math.Soc.. (発表予定).

Kazunori Kodaka：“与 2-环面的 Furstenberg 变换相关的 Ko 群交叉乘积的正锥体”Proc.Edinburgh Math.Soc..（待提交）。

Yuu Hariya: "Diffusion processes on path Spaces with interactions"Rev.Math.Phys.. (to appear).

Yuu Hariya：“具有相互作用的路径空间上的扩散过程”Rev.Math.Phys..（即将出现）。