Research of stochastic processes on fractals.

1.我们得到了p.c.f.上扩散过程的跃迁密度（热核）的精确估计。自相似集，它对应于自相似分形。已知如果分形具有强对称性，那么布朗运动的热核具有Aronson型估计。在我们的结果中，我们证明了Aronson型估计一般不成立。这项工作将出现在J.伦敦Math.Soc.2。在无穷分歧分形上，我们研究了随机Sierpinski地毯上扩散过程的热核估计。我们获得了每个样品地毯（每个环境）的精确估计。此外，我们得到了几乎肯定的估计假设强遍历地毯的随机性。其核心思想之一是利用Barlow-Bass的耦合变元得到近似过程的一致Harnack不等式。这件作品现在是预印本。在分形与欧氏空间的关系上，研究了均匀化问题。自从首席研究员与Kusuoka教授的联合工作以来，人们知道重整化映射不动点的稳定性至关重要。在研究过程中，我们通过参加国际研讨会，与同领域的研究人员进行了讨论，学到了一些新的思路和方法来探索这个问题。但到目前为止，我们还不能将这些方法应用于我们的案例。这是我们在不久的将来应该解决的问题。

1. We have obtained sharp estimates on the transition densities (heat kernels) for diffusion processes on p.c.f. self-similar sets, which correspond to finitely ramified self-similar fractals. It was known that if the fractal had a strong symmetry, then the heat kernel of the Brownian motion had Aronson type estimates. In our result, we show that the Aronson type estimates do not hold in general. This work will appear in J.London Math.Soc.2. On infinitely ramified fractals, we have studied the heat kernel estimates for diffusion processes on random Sierpinski carpets. We obtained sharp esimates for each sample carpets (each environments). Further, we obtained almost sure estimates assuming strong ergodicity for the randomness of the carpets. One of the key idea was to obtain uniform Harnack inequality of the approximate processes using the coupling arguments due to Barlow-Bass. This work is now a preprint.3. On the relations between fractals and Euclidean spaces, we studied homogenization problems. Since the joint work of the head investigator with Prof.Kusuoka, it was known that the stability of fixed points of the renormalization map was essential. In our research, we discussed with researchers of the same fields when we attended interational workshops and learned several new ideas and methods to search for the problem. But so far we could not apply the methods to our cases. This is the problem we should pursue in a near future.