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 …

The main theme of this book is the interplay between the behaviour of a class of stochastic
processes (random walks) and discrete structure theory. The author considers Markov …

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in
bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We …

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity)
and the notion of a fractal dimension D which exceeds the topological dimension d. In this …

The construction of brownian motion on the Sierpinski carpet Page 1 ANNALES DE L’IHP,
SECTION B MARTIN T. BARLOW RICHARD F. BASS The construction of brownian motion …

Estimates of heat kernels on Riemannian manifolds Page 154 Estimates of heat kernels on
Riemannian manifolds Alexander Grigor'yan Imperial College London SW7 2BZ England email …

TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES Page 1 The
Annals of Probability 2012, Vol. 40, No. 3, 1212-1284 DOI: 10.1214/11-AOP645 © Institute of …

We consider a metric measure space $(M, d,\mu) $ and a heat kernel $ p_ {t}(x, y) $ on $ M $
satisfying certain upper and lower estimates, which depend on two parameters $\alpha …

Assume that there is some analytic structure, a differential equation or a stochastic process
for example, on a metric space. To describe asymptotic behaviors of analytic objects, the …

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a
generalized Sierpinski carpet that is invariant with respect to the local symmetries of the …