Here are some details.

The main theme of these lectures is to analyze heat conduction on disordered
media such as fractals and percolation clusters using both

probabilistic and analytic methods,
and to study the scaling limits of Markov chains on the media.

The problem of random walk on a percolation cluster `the ant in the labyrinth' has received
much attention both in the physics and the

mathematics literature. In 1986, H. Kesten showed
an anomalous behavior of a random walk on a percolation cluster at critical probability

for trees
and for *Z*^{2}. (To be precise, the critical percolation cluster is finite, so the random walk
is considered on an incipient infinite cluster

(IIC), namely a critical percolation cluster conditioned
to be infinite.) Partly motivated by this work, analysis and diffusion processes on fractals

have been
developed since the late eighties. As a result, various new methods have been produced to estimate
heat kernels on disordered media,

and these turn out to be useful to establish quenched estimates
on random media. Recently, it has been proved that random walks on IICs are

sub-diffusive
on *Z ^{d}* when

Throughout the lectures, I will survey the above mentioned developments in a compact way.
In the first part of the lectures, I will summarize

some classical and non-classical estimates for
heat kernels, and discuss stability of the estimates under perturbations of operators and spaces.

Here Nash inequalities and equivalent inequalities will play a central role.
In the latter part of the lectures, I will give various examples of

disordered media and obtain heat kernel estimates for Markov chains on them. In some models, I will also discuss
scaling limits of the Markov

chains. Examples of disordered media include fractals, percolation
clusters, random conductance models and random graphs.

Lecture Notes (Version for St. Flour Lectures) PDF File (887kb)
Corrections (19 November) PDF File

Now published from LNM, Springer!

T. Kumagai, Random Walks on Disordered Media and their Scaling Limits.

Lecture Notes in Mathematics, Vol. 2101,
École d'Été de Probabilités de Saint-Flour XL--2010. Springer, New York, (2014). Corrections

The titles of my lectures are as follows.

(1) Parabolic Harnack inequalities and heat kernel estimates for stable-like processes on metric measure spaces

(2) Convergence of symmetric Markov chains on *Z ^{d}
*PDF File

My lectures are closely related to the lectures by Z.-Q. Chen. In fact, the whole 4 lectures cover many of the recent
developments

on heat kernel
estimates and related topics on jump processes.
For the reference, the titles of Chen's lectures are as follows:

(A) Heat kernel estimates for jump diffusions

(B) Heat kernel estimates for stable processes in open sets
A HREF="BBKjp.pdf">PDF File (208kb)

The following 4 papers are most relevant to my lectures.

Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps.

Math. Z. 261 (2009), no. 2, 297--320. PDF File (208kb), Post Script File. (458kb)

Heat kernel upper bounds for jump processes and the first exit time.

J. Reine Angew. Math. 626 (2009), 135--157. PDF File (318kb), Post Script File. (779kb) Corrections PDF File (104Kb)

Convergence of symmetric Markov chains on Z

Probab. Theory Relat. Fields, 148 (2010), 107--140. (Revised Version) PDF File (252kb) PS File (552kb) Correction PDF File (68Kb)

Symmetric Markov chains on Z

Trans. Amer. Math. Soc., 360 (2008), no. 4, 2041--2075. PDF File (525kb), Post Script File. (558kb)

Other related papers are
52), 49), 48), 44), 43), 37), 36), 27), 23), 22) in this page.