Recent developments of analysis on fractals

“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