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

我们完成并发表了一篇论文，指出在多维空间中，即使粒子的密度很高，具有凸硬核的粒子的自扩散系数也是正的。为了证明这一点，我们使用了自扩散系数的变分公式和由定向位渗流的一个结果得到的Gibbs测度的精细估计。相互作用的布朗运动是无限数量的布朗粒子与相互作用的运动的动力学。构造这样的动力学有一些困难，因为这确实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发表了这篇论文，比如系数是可测函数。到目前为止，结果只知道系数是上半连续的限制假设。我们对此进行了改进，使它们由上半连续函数或上半连续函数从下到上都有界。我们认为这种推广是相当令人满意的。我们证明了同一位置的两个粒子的存在能力的正性对于一维自扩散系数的正性是必要的。虽然我们试图证明这也是充分的，但这是徒劳的。在做这项研究的同时，我们来到了新的主题，使得在无限体积的路径空间中同样的问题。它与Log Sobolev不等式有关；所以我现在认为这个新问题很令人兴奋。

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.