Analysis on fractals

Summary

Analysis on fractals builds Laplacians, energies, harmonic functions, heat equations and diffusion processes on fractal sets such as the Sierpiński gasket. On such sets there are no smooth curves or tangent vectors, so the Laplacian cannot be defined by derivatives. Instead one takes limits of renormalized energies and Laplacian matrices of finite graphs that approximate the fractal, as in Kigami's construction. Another route builds Brownian motion on the fractal as a limit of random walks on these graphs. The key tools are Dirichlet forms and resistance forms. The results show new features, such as a spectral dimension that differs from the Hausdorff dimension.