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

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。我们以这样一种方式改进了这一点，即它们从上到下都是由上连续函数所限定的。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。

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.