The art of random walks

András Telcs (auth.)9783540330271, 3540330275

Einstein proved that the mean square displacement of Brownian motion is proportional to time. He also proved that the diffusion constant depends on the mass and on the conductivity (sometimes referred to Einstein’s relation). The main aim of this book is to reveal similar connections between the physical and geometric properties of space and diffusion. This is done in the context of random walks in the absence of algebraic structure, local or global spatial symmetry or self-similarity. The author studies the heat diffusion at this general level and discusses the following topics:

The multiplicative Einstein relation,

Isoperimetric inequalities,

Heat kernel estimates

Elliptic and parabolic Harnack inequality.

Table of contents :
Front Matter….Pages i-vii
Front Matter….Pages 24-24
Introduction….Pages 1-6
Basic definitions and preliminaries….Pages 7-21
Some elements of potential theory….Pages 25-47
Isoperimetric inequalities….Pages 49-60
Polynomial volume growth….Pages 61-67
Front Matter….Pages 70-70
Motivation of the local approach….Pages 71-81
Einstein relation….Pages 83-93
Upper estimates….Pages 95-129
Lower estimates….Pages 131-151
Two-sided estimates….Pages 153-163
Closing remarks….Pages 165-168
Parabolic Harnack inequality….Pages 169-179
Semi-local theory….Pages 181-185
Back Matter….Pages 187-199