The notes are based on lectures given in St. Flour in 1995, and cover, in greater detail, most
of the course given there. The word" fractal" was coined by Mandelbrot [Man] in the 1970s …

This book covers analysis on fractals, a developing area of mathematics which focuses on
the dynamical aspects of fractals, such as heat diffusion on fractals and the vibration of a …

We consider a class of fractal subsets of formed in a manner analogous to the construction
of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic …

Upper and lower bounds are obtained for the transition densities p (t, x, y) of Brownian
motion on the Sierpinski carpet. These are of the same form as those which hold for the …

Let Fn be the n th stage in the construction of the Sierpiński carpet. Let Rn be the electrical
resistance of Fn when the left and right sides are each short-circuited, and a voltage is …

Abstract Let (𝒳, d, m) be a metric measure space with a local regular Dirichlet form. We
establish necessary and sufficient conditions for upper heat kernel bounds with sub-diffusive …

Non-Gaussian upper and lower bounds are obtained for the transition probabilities of the
simple random walk on the Sierpinski graph, the pre-fractal associated with the Sierpinski …

In this article, local limit theorems for sequences of simple random walks on graphs are
established. The results formulated are motivated by a variety of random graph models, and …

We show that the effective diffusivity matrix D (V n) for the heat operator∂ t−(Δ/2−∇ V n∇) in
a periodic potential V n= Σ math image U k (x/R k) obtained as a superposition of Hölder …

We consider homogeneous random Sierpinski carpets, a class of in® nitely rami® ed
random fractals which have spatial symmetry but which do not have exact self-similarity. For …