Analysis on Metric Spaces and Walk Dimension

Explore the fundamental concepts of Analysis on Ahlfors-regular metric spaces, with a focus on fractals, in this comprehensive lecture. Delve into the two key parameters that characterize these spaces: the Hausdorff dimension and the walk dimension. Examine how the Hausdorff dimension influences the Hausdorff measure and integral calculus on these spaces. Investigate the walk dimension, a novel invariant of regular metric spaces, and its significance in determining space/time scaling for diffusion processes. Learn about the connection between the walk dimension and the Laplace operator analogue, and how it impacts differential calculus on the underlying space. Discover how the walk dimension can be defined for any regular metric space using a critical exponent of Besov function spaces. Compare the walk dimensions of various spaces, including Euclidean spaces, fractal spaces, and ultra-metric spaces, to gain a deeper understanding of their analytical properties.