TITLE OF PROJECT : SELF-DIFFUSION MATRIX OF INTERACTING BROWNIAN MOTIONS

项目名称：相互作用布朗运动的自扩散矩阵

• 批准号: 09640244
• 负责人: OSADA Hirofumi
• 金额: $ 1.92万
• 依托单位: NAGOYA UNIVERSITY (1998)The University of Tokyo (1997)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640244/
• 关键词: SELF=DIFFUSION COEFFICIENTS; INFINITE PARTICLE SYSTEM; INTERACTING BROWNIAN MOTION; HYDRODYNAMICAL LIMIT

We completed and published the paper that says the self-diffusion coefficient is positive for particles with convex hard cores in a multi-dimensional space, even if the density of the particle is very high. In order to prove this we use the variational formula of the self-diffusion coefficient and a fine estimate of Gibbs measures derived from a result on oriented site percolation.Interacting Brownian motion is a dynamics of the motion of infinite amount of Brownian particles with interaction. To construct such a dynamics has some difficulty because this is indeed a problem to construct infinitely dimensional diffusion. For this we have solved the problem and publised the paper under very mild assumption such as the coefficients are measurable functions. So far the results are known only the restrict assumption such that the coefficients are upper semicontinuous. We improve this in such a way that they are bounded from both of below and above by upper semicontinuous functions. We think this generalization is quite satisfactory.We prove the positivity of the capacity of the existence of two particles at the same position is necessary for the positivity of the one dimensional self-diffusion coefficient. Althogh we tried to prove this is also sufficient, it is in vain.While doing this research, we come to the new thema such that the same problem in the infinite volume path space. It is related to Log Sobolev inequality ; so I am now think this new problem is exciting.

我们完成并发表了该论文，该论文说，即使粒子的密度非常高，在多维空间中具有凸硬芯的粒子的自扩散系数是阳性的。为了证明这一点，我们使用自扩散系数的变异公式，以及从定向位置渗透的结果得出的Gibbs测量的良好估计值。互动的布朗运动是无限量的布朗尼与相互作用的运动动态的动力学。构建这种动力学有一定的困难，因为这确实是构建无限维扩散的问题。为此，我们解决了问题，并在非常温和的假设（例如系数）下发表了论文。到目前为止，结果仅是限制假设，因此系数是上半连续的。我们以这样的方式改进了这一点，以使它们从下层和更高的半连续函数中界定。我们认为这种概括是非常令人满意的。我们证明，对于一个维度自扩散系数的积极性，必须证明存在两个粒子的能力的阳性。 Althogh我们试图证明这也足够了，这是徒劳的。尽管进行这项研究，但我们来到了新的主题，因此在无限的体积路径空间中存在相同的问题。它与Log Sobolev不平等有关；因此，我现在认为这个新问题令人兴奋。

项目成果

長田博文: "An invariance principle for Markov processes and Brownian particles with singular interaction" Ann.Inst.Henri Poincare,Probabilites et Statistiques. 34-n^○2. 217-248 (1998)

Hirofumi Nagata：“马尔可夫过程和布朗粒子与奇异相互作用的不变性原理”Ann.Inst.Henri Poincare，Probabilites et Statistiques 34-n^○2（1998）。

舟木直久: "相分離の確率モデルと界面の運動方程式" 数学. 50-1. 68-85 (1998)

Naohisa Funaki：“相分离的随机模型和界面运动方程”50-1（1998）。

Hirofumi Osada: "{\it Interacting Brownian particles with measurable potentials}" Proc.\ Japan Acad.{\bf 74}, Ser.\ A. 10-12 (1998)

Hirofumi Osada：“{它与可测量势能的布朗粒子相互作用}”Proc. Japan Acad.{f 74}，Ser. A. 10-12 (1998)

長田博文: "Interacting Brownian motions with measurable potentials" Proceedings of the Japan Academy. 74-A. 10-12 (1998)

Hirofumi Nagata：“布朗运动与可测量势的相互作用”日本科学院院刊 74-A。