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.

我们完成并发表了一篇论文，该论文说，对于多维空间中具有凸硬核的粒子，自扩散系数是正的，即使粒子的密度非常高。为了证明这一点，我们利用自扩散系数的变分公式和由定向位渗流的结果导出的Gibbs测度的精细估计.相互作用布朗运动是无限数量的布朗粒子在相互作用下的运动动力学.构造这样的动力学有一定的困难，因为这实际上是一个构造无限维扩散的问题。为此，我们解决了这个问题，并在非常温和的假设下，如系数是可测的功能，使文件。到目前为止，结果只知道限制的假设，使系数是上连续的。我们对此进行了改进，使它们从下到上都受到上半连续函数的约束。我们认为这种推广是令人满意的，证明了两个粒子在同一位置存在的能力的正性是一维自扩散系数的正性的必要条件。虽然我们试图证明这也是充分的，但这是徒劳的，在做这项研究时，我们得到了新的主题，即在无限体积的道路空间中同样的问题。它与Log Sobolev不等式有关，所以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）。

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。

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)