Recent developments of analysis on fractals
Recent developments of analysis on fractals
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DOI：
10.1090/trans2/223/06
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
T. Kumagai
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文献类型：
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作者：
T. Kumagai
“Fractal”是法国数学家B·B·曼德尔布罗特在20世纪70年代左右发明的一个词。他声称，自然界的许多模式，如云、山和海岸线，都不是光滑的线条或圆形，而是如此不规则、零散，呈现出完全不同的复杂程度。从这个角度来看，他将这些形状的家族称为分形族。特别是，他特别注意了分形的自相似性质，即全局部分和局部部分之间的相似性。大约在同一时间，数学物理学家开始利用无序介质的一些自相似特性来分析无序介质的特性，例如聚合物和网络结构、模具和晶体的生长(见[30，31])。在这些工作的激励下，数学家们对分形学的分析性质产生了兴趣，如热传递和波传递。分形是无序介质的典型的理想例子。由于分数不是光滑性的，所以人们不能直接定义微分的概念。因此，最大的问题是如何严格地对待这种物理现象。八十年代中期，概率学家通过在Sierpinski垫片上构造一个扩散过程来解决这个问题，这是一个典型的分形。第一个工作是Goldstein([21])和Kusuoka([50])；Barlow-Perkins([11])进一步得到了扩散的详细热核估计，我们将在后面更详细地讨论这一估计。之后，Kigami([41])在垫片上构造了一个Laplace算子作为差分算子的极限。这种分析方法激励了Fukushima-Shima([18])的一项工作，该工作清楚地表明Dirichlet形式理论非常适用于这一领域。从那时起，在过去的几十年里，扩散过程(以及相应的自伴算子)被构造在各种类型的分形图上，并对其性质进行了深入的研究。现在很清楚，分数维上的扩散与欧几里得空间上的扩散具有完全不同的性质。例如，可以理解，这样的过程通常具有次扩散行为，并且布朗运动的热核在“好的”分形图上享受亚高斯估计(见(2.2))。通过大量的工作，关于分形学的随机过程已经涉及到其他各个领域。近年来，这一领域有了新的发展，即以分析类分形空间为目标，而不是
“Fractal” is a word invented by French mathematician B.B. Mandelbrot around 1970s. He claimed that many patterns of Nature, such as clouds, mountains and coastlines are not lines nor circles which are smooth, but are so irregular, fragmented and exhibit an altogether different level of complexity. From this viewpoint, he called the family of those shapes as fractals. Especially, he made special attention to the self-similar property, i.e. similarity between global parts and local parts, of fractals. Around the same time, mathematical physicists began to analyse properties of disordered media such as the structure of polymers and networks, growth of molds and crystals, using some self-similar properties of the media (see, for instance, [30, 31]). Motivated by these works, mathematicians got interested in the analytical properties of fractals such as heat transfer and wave transfer. Fractals are typical ideal examples of the disordered media. Since there is no smoothness on fractals, one cannot define the notion of differentials directly. So the biggest problem was how to treat such physical phenomena in a rigorous way. In the middle eighties, probabilists solved the problem by constructing a diffusion process on the Sierpinski gasket, which is a typical fractal. The first works are by Goldstein ([21]) and Kusuoka ([50]); Barlow-Perkins ([11]) further obtained detailed heat kernel estimates of the diffusion which we will discuss in more details later. After that, Kigami ([41]) constructed a Laplace operator on the gasket as a limit of difference operators. This analytical approach motivated a work by Fukushima-Shima ([18]), which made it clear that the theory of Dirichlet forms was well-applicable to this area. Since then, for the last several decades, diffusion processes (and the corresponding self-adjoint operators) have been constructed on various classes of fractals and their properties have been deeply studied. It is now getting clear that the diffusions on fractals have completely different properties from diffusions on Euclidean spaces. For instance, it is understood that such processes typically have sub-diffusive behaviour and heat kernels for Brownian motion on ‘nice’ fractals enjoy sub-Gaussian estimates (see (2.2)). Through substantial amount of work, stochastic processes on fractals have been related to various other fields. Recently, there are new developments of this area, namely to aim for analysis on “fractal-like spaces” instead of