Diffusions and spectral analysis on fractals

The classical diffusion process was first studied by Einstein, and later a mathematical theory was developed by Wiener, Kolmogorov, Levy et al. One of the basic principle is that displacement in a small time is proportional to the square root of time. Recently it was discovered that some fractals carry natural diffusion processes that obey scaling laws what are different from the classical Gaussian diffusion, but are of so called sub-Gaussian type. Moreover, in some situations the diffusion, and therefore the correspondent Laplace operator, are uniquely determined by the geometry of the space. Recently it was proved for Sierpinski square and its generalizations (a joint work with M. T. Barlow, R. F. Bass, T. Kumagai). In the second half of the talk spectral analysis of the intrinsic Laplacian on finitely ramified fractals with symmetries will be discussed.