Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets
Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets
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DOI：
10.2969/jmsj/05220373
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发表时间：
2000-04
影响因子：
0.7
通讯作者：
B. Hambly;T. Kumagai;S. Kusuoka;X. Zhou
中科院分区：
文献类型：
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作者：
B. Hambly;T. Kumagai;S. Kusuoka;X. Zhou
我们考虑齐次随机Sierpinski地毯，这是一类具有空间对称性但不具有精确自相似性的随机分形。对于混合环境，我们在分形上构建“自然”扩散过程，并获得该过程的过渡密度的上限和下限估计，该过程最高可能达到常数。考虑随机情况，当环境平稳且遍历时，我们推导出了Aronson型估计。1. 介绍。Sierpinski地毯是R de的分形子集，作为八个收缩映射族的混合点。我们可以等价地构造分形：取；1、将它分成9个等边长为1/3的正方形，去掉中间的正方形。然后对剩下的8个方格重复此过程，并逐次迭代。地毯是由此产生的分形，具有Hausdor维数df ø log 8=log 3。这个集合的一个基本几何性质是它的非均匀性，因为分形的任何连通子集只能通过移除维数为1的集合而与其他子集断开。这使得对这个集合的分析比Sierpinski垫圈的情况更加di1 / 2，（由将三角形划分为四个等面积三角形形成的集合，反复去除中心，向下指向的三角形），这是一个完全的rami®ed集合，在该集合中只需要去除一个分形的子集。之前的研究主要集中在具有精确自相似性的广义Sierpinski地毯上。在[3]，[4],[5]，[6]系列论文中，确定了二维地毯上各向同性扩散过程布朗运动的存在性及其性质。这一过程被确定为1991年数学学科分类教学的弱极限。主要60 j60;次级60B05、60J35。关键词和短语。Sierpinski地毯，随机分形，扩散过程，热方程，过渡密度。*这项研究得到了惠普公司BRIMS皇家学会工业奖学金的部分支持。**由JSPS计划部分支持的研究。收敛于分形的R子集序列上的一系列反射布朗运动。使用这种概率方法，可以检查分形上的拉普拉斯核和热核，因为它们分别是布朗运动的极小发生器和跃迁密度。证明这些对象存在的关键在于建立一个哈纳克不等式，这是通过一个简单的二维耦合论证来实现的。在b[8]中，这项工作被推广到更高维度的地毯上，使用一个更复杂的耦合论证来证明必要的哈纳克不等式。我们将在这里关注一类任何尺寸的Sierpinski地毯，但增加了尺度不规则的特征。目前已有许多关于非自相似分形的结果，[11],[18],[19]。我们已经考虑了两种自然的“随机”分形。一是空间均匀但尺度不规则，二是空间不对称。对于这些分形，在热核中有比在完全自相似情况下观察到的更大的振荡。我们将考虑一类尺度不规则的随机分形，从而扩展了[18]，[11]中开始的齐次随机分形的工作。我们不考虑随机递归分形[19]，因为它对我们通过哈纳克不等式的方法至关重要，我们类中的分形具有空间同质性。我们构造了一个简单的分形例子，我们将在本文中考虑。首先定义一组二维地毯，对于nv3，我们称之为SC(n)，其中地毯的边长除以n，去掉边长为ny²=n的中心正方形。这样就得到了一个Hausdor维数为df ø log 4 ny 1 * =log n的地毯族，其中n ø 3为第一个成员，即原来的Sierpinski地毯，它与情况n ø 4如图1所示。为了构造一个尺度不规则的地毯，我们取一个序列，其中xn a f3；4g, En，称为环境序列。然后，我们根据序列应用对应于类型3或类型4的a1 / 2ne变换。图1：来自SC(n) B. M. Hambly、T. Kumagai、S. Kusuoka和X. Y. Zhou 374家族的两条Sierpinski地毯
We consider homogeneous random Sierpinski carpets, a class of in®nitely rami®ed random fractals which have spatial symmetry but which do not have exact self-similarity. For a ®xed environment we construct ``natural'' di usion processes on the fractal and obtain upper and lower estimates of the transition density for the process that are up to constants best possible. By considering the random case, when the environment is stationary and ergodic, we deduce estimates of Aronson type. 1. Introduction. The Sierpinski carpet is a fractal subset of R de®ned as the ®xed point of a family of eight contraction maps. We can equivalently construct the fractal by taking 0; 1, dividing it into nine equal squares of side length 1/3, and removing the central square. This procedure is then repeated for each of the eight remaining squares and iterated in®nitely. The carpet is the resulting fractal and has Hausdor dimension df log 8=log 3. A fundamental geometrical property of this set is its in®nite rami®cation, in that any connected subset of the fractal can only be disconnected from the rest by removing a set of dimension 1. This makes analysis on this set much more di1⁄2cult than for the case of the Sierpinski gasket, (the set formed from dividing a triangle into four equal area triangles with repeated removal of the central, downward pointing triangle) which is a ®nitely rami®ed set in that removal of only a ®nite number of points is required to disconnect a subset of the fractal. The previous work on in®nitely rami®ed fractals has concentrated on generalised Sierpinski carpets with exact self-similarity. In a series of papers [3], [4], [5], [6 ], the existence and properties of a Brownian motion, an isotropic di usion process, on the two dimensional carpet were determined. This process was de®ned as the weak limit of 1991 Mathematics Subject Classi®cation. Primary 60J60; Secondary 60B05, 60J35. Key Words and Phrases. Sierpinski carpet, random fractal, di usion process, heat equation, transition densities. * Research partly supported by a Royal Society Industry Fellowship at BRIMS, Hewlett-Packard. ** Research partly supported by JSPS Program. a sequence of re ected Brownian motions on a sequence of subsets of R converging to the fractal. Using this probabilistic approach it is possible to examine the Laplacian and the heat kernel on the fractal as these are respectively, the in®nitesimal generator and transition density of the Brownian motion. The key to proving the existence of these objects lies in establishing a Harnack inequality, which is accomplished via a straightforward coupling argument in two dimensions. In [8], this work was extended to higher dimensional carpets, using a more complicated coupling argument to prove the necessary Harnack inequality. We will be concerned here with a class of Sierpinski carpets in any dimension but with the added feature of scale irregularity. There have now been many results on ®nitely rami®ed fractals and in this setting some non-self-similar sets have been explored, [11], [18], [19]. There are two natural `random' fractals that have been considered. Firstly one with spatial homogeneity but scale irregularity and secondly, one without spatial symmetry. For these fractals there are greater oscillations in the heat kernel than that observed in the exactly self-similar case. We will consider a class of in®nitely rami®ed fractals which are scale irregular, thus extending the work on homogeneous random fractals initiated in [18], [11]. We do not consider random recursive fractals [19], as it is essential to our approach via the Harnack inequality, that there is spatial homogeneity for the fractals in our class. We construct a simple example of the fractals that we will consider in this paper. Firstly de®ne a family of two dimensional carpets, which we will call SC(n) for nV 3, where the side length of the carpet is divided by n and a central square of side ny 2=n is removed. This gives a family of carpets of Hausdor dimension df log 4 ny 1=log n, the ®rst member, with n 3 is the original Sierpinski carpet and it, along with the case n 4, is shown in Figure 1. In order to construct a carpet with scale irregularity we take a sequence fxng y n1 where xn A f3; 4g, En, called the environment sequence. We then apply the a1⁄2ne transformations corresponding to either type 3 or 4 according to the sequence. In this Figure 1: Two Sierpinski carpets from the family SC(n) B. M. Hambly, T. Kumagai, S. Kusuoka and X. Y. Zhou 374