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 Z2. (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 Zd when d is high enough, on trees, and on the spread-out oriented percolation for d>6.
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 Zd 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
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.
Other related papers are
52), 49), 48), 44), 43), 37), 36), 27), 23), 22) in this page.
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