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Heat Kernel Estimates for Symmetric Random Walks on a Class of Fractal Graphs and Stability Under Rough Isometries

2002 Fractals
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Release : 2002
Genre : Fractals
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Heat Kernel Estimates for Symmetric Random Walks on a Class of Fractal Graphs and Stability Under Rough Isometries by : Ben M. Hambly

Download or read book Heat Kernel Estimates for Symmetric Random Walks on a Class of Fractal Graphs and Stability Under Rough Isometries written by Ben M. Hambly. This book was released on 2002. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We examine a class of fractal graphs which arise from a subclass of finitely ramified fractals. The two-sided heat kernel estimates for these graphs are obtained in terms of an effective resistance metric and they are best possible up to constants. If the graph has symmetry, these estimates can be expressed as the usual Gaussian or sub-Gaussian estimates. However, without symmetry, the off-diagonal terms show different decay in different directions. We also discuss the stability of the sub-Gaussian heat kernel estimates under rough isometries."

The Art of Random Walks

2006-05-17 Mathematics
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Release : 2006-05-17
Genre : Mathematics
Kind : eBook
Book Rating : 275/5 ( reviews)

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Book Synopsis The Art of Random Walks by : Andras Telcs

Download or read book The Art of Random Walks written by Andras Telcs. This book was released on 2006-05-17. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces

2003 Mathematics
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Release : 2003
Genre : Mathematics
Kind : eBook
Book Rating : 839/5 ( reviews)

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Book Synopsis Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces by : Pascal Auscher

Download or read book Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces written by Pascal Auscher. This book was released on 2003. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the expanded lecture notes of courses taught at the Emile Borel Centre of the Henri Poincare Institute (Paris). In the book, leading experts introduce recent research in their fields. The unifying theme is the study of heat kernels in various situations using related geometric and analytic tools. Topics include analysis of complex-coefficient elliptic operators, diffusions on fractals and on infinite-dimensional groups, heat kernel and isoperimetry on Riemannian manifolds, heat kernels and infinite dimensional analysis, diffusions and Sobolev-type spaces on metric spaces, quasi-regular mappings and $p$-Laplace operators, heat kernel and spherical inversion on $SL 2(C)$, random walks and spectral geometry on crystal lattices, isoperimetric and isocapacitary inequalities, and generating function techniques for random walks on graphs. This volume is suitable for graduate students and research mathematicians interested in random processes and analysis on manifolds.

Random Walks on Disordered Media and their Scaling Limits

2014-01-25 Mathematics
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Release : 2014-01-25
Genre : Mathematics
Kind : eBook
Book Rating : 52X/5 ( reviews)

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Book Synopsis Random Walks on Disordered Media and their Scaling Limits by : Takashi Kumagai

Download or read book Random Walks on Disordered Media and their Scaling Limits written by Takashi Kumagai. This book was released on 2014-01-25. Available in PDF, EPUB and Kindle. Book excerpt: In these lecture notes, we will analyze the behavior of random walk on disordered media by means of both probabilistic and analytic methods, and will study the scaling limits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. The first few chapters of the notes can be used as an introduction to discrete potential theory. Recently, there has been significant progress on the theory of random walk on disordered media such as fractals and random media. Random walk on a percolation cluster(‘the ant in the labyrinth’)is one of the typical examples. In 1986, H. Kesten showed the anomalous behavior of a random walk on a percolation cluster at critical probability. 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. These developments are summarized in the notes.

Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot

2004 Mathematics
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Release : 2004
Genre : Mathematics
Kind : eBook
Book Rating : 382/5 ( reviews)

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Book Synopsis Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot by : Michel Laurent Lapidus

Download or read book Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot written by Michel Laurent Lapidus. This book was released on 2004. Available in PDF, EPUB and Kindle. Book excerpt: This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry. In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications. This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.

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