Research of stochastic processes on fractals.

分形随机过程的研究。

• 批准号: 08454040
• 负责人: KUMAGAI Takashi
• 金额: $ 1.15万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454040/
• 关键词: fractal; stochastic process; self-similar set; heat kernel; sierpinski carpet; Harnack ineguality; homogenization; 拡散過程; ランダムウォーク; ハウスドルク次元; 固有値分布

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.

1.我们得到了P.C.F上扩散过程转移密度(热核)的精确估计。自相似集，它对应于有限分支的自相似分形图。已知如果分数维具有强对称性，则布朗运动的热核具有Aronson型估计。在我们的结果中，我们表明Aronson型估计在一般情况下不成立。这项研究将发表在《数学杂志》上。在无限分枝分形上，我们研究了随机Sierpinski地毯上扩散过程的热核估计。我们为每个样品地毯(每个环境)获得了锐利的esimate。进一步，我们得到了几乎肯定的估计，假设地毯的随机性具有很强的遍历性。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。这本书现在是预印本。在分数维与欧氏空间之间的关系上，我们研究了齐化问题。自从首席研究员与Kusuoka教授共同工作以来，人们知道重整化映射的不动点的稳定性是必不可少的。在我们的研究中，我们在参加国际研讨会时与相同领域的研究人员进行了讨论，并学习了几种寻找问题的新思路和方法。但到目前为止，我们还不能将这些方法应用到我们的案例中。这是我们在不久的将来应该追求的问题。

项目成果

N.Obata: "Quantum stochastic differential equations in terms of quantum white noise" Nonlinear Anal-ysis Theory, Methods and Applications. 30. 279-290 (1997)

N.Obata：“量子白噪声方面的量子随机微分方程”非线性分析理论、方法和应用。

尾畑伸明: "Wick product of white noise operators and quantum stochastic differential equations." J.Math.Soc.Japan. 51(発表予定).

Nobuaki Obata：“白噪声算子和量子随机微分方程的 Wick 乘积。”J.Math.Soc.Japan 51（待提交）。

尾畑 伸明: "Quantum stochastic differential equations in terms of quantum white noise" Nonlinear Analysis,Theory,Methods and Applications. 30. 279-290 (1997)

Nobuaki Obata：“量子白噪声方面的量子随机微分方程”非线性分析、理论、方法和应用 30. 279-290 (1997)。

T.Kumagai: "Stochactic processes on fractals and their related fields." Sugaku. 49-2 (in Japanese). 158-172 (1997)

T.Kumagai：“分形及其相关领域的随机过程。”

市原完治: "A Liouville property for difference operators" Japanese J.Math.(発表予定).

Kanji Ichihara：“差分算子的刘维尔性质”日本 J.Math（待提交）。