The Art of Random Walks

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:

1. The multiplicative Einstein relation,
2. Isoperimetric inequalities,
3. Heat kernel estimates
4. Elliptic and parabolic Harnack inequality.

• Brownian motion
• diffusion
• heat kernel
• isoperimetric inequalities
• random walk
• reversible Markov chain
• partial differential equations

From the reviews:

"This book studies random walks on countable infinite connected weighted graphs, with particular emphasis on fractal graphs like the Sierpinski triangular graph or the weighted Vicsek tree. … The book is intended to be self-contained and accessible to graduate and Ph.D. students. It contains a wealth of references, also on various aspects of random walks not covered by the text." (Wolfgang König, Mathematical Reviews, Issue 2007 d)

"This book studies random walks on countable infinite connected weighted graphs, with particular emphasis on fractal graphs like the Sierpinski triangular graph or the weighted Vicsek tree. … The book is intended to be self-contained and accessible to graduate and PhD students. It contains a wealth of references, also on various aspects of random walks not covered by the text. At the end of the book a list of some dozens of types of inequalities appear that are introduced in the book" (Wolfgang König, Zentralblatt MATH, Vol. 1104 (6), 2007)

• Department of Computer Science and Information Theory, Budapest University of Technology, Electrical Engineering and Informatics, 1117, Budapest, Hungary

  András Telcs

András Telcs is associated professor of the Budapest University of Technology. Formerly he taught statistics in business schools as well as worked for major libraries. His main research interests are random walks, discrete potential theory, active on different application of probability and statistics.

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