Diffusion processes on Mandala
A mathematical representation of fractals were given by \textit{J. E. Hutchinson} [Indiana Univ. Math. J. 30, 713-747 (1981; Zbl 0598.28011)]. The study of diffusion processes on fractals was initiated by \textit{S. Kusuoka} [in: Probabilistic methods in mathematical physics, 251-274 (1987; Zbl 0645.60081)], and others. Hutchinson's fractal \(K\) is associated with full-shift symbolic space. The author studies a finitely ramified fractal which is not associated with full-shift, but with a Markov sub-shift. Let \(C\) be a unite circle in \(\mathbb{R}^2\) with the origin as its center, and a collection \(\{F_1, \dots, F_5\}\) of 3-similitudes with five points. In this case, there exists a unique compact set \(K\subset \mathbb{R}^2\) such that \(K= \bigcup^5_{i=1} F_i(K) \cup C\), cf. \textit{M. Hata} (1985), which is called the Mandala. A mathematical definition of the plain Mandala is presented in Section 2. The method of constructing diffusion processes is a modification on Kigami's method for past critically finite selfsimilar sets [see \textit{J. Kigami}, Trans. Am. Math. Soc. 335, No. 2, 721-755 (1993; Zbl 0773.31009) and \textit{T. Kumagai}, J. Math. Kyoto Univ. 33, No. 3, 765-786 (1993; Zbl 0798.58042)]. The plain Mandala is denoted by \(K\). In Section 3 there is defined a difference operator on \(H_m\) on the space \(\ell(V_m)\) and there is introduced a bilinear form. In Section 4, under Condition A there is considered a measure \(\mu\) of Bernoulli type. In Section 5, under condition \(B\), a result is proved on functions of the plain Mandala \(K\). In Section 6, under condition \(A\), it is determined the spectral dimension of the diffusion process on the plain Mandala. In Section 7 there are studied to what extent the arguments in Sections 4 and 6 work.
scientific article; zbMATH DE number 902838