The Art of Random WalksEinstein 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:
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Contents
| 1 | |
| 7 | |
3 | 25 |
Isoperimetric inequalities 49 | 48 |
Polynomial volume growth | 61 |
6 | 69 |
Einstein relation | 83 |
Upper estimates 95 | 94 |
Lower estimates | 131 |
Twosided estimates 153 | 152 |
Parabolic Harnack inequality | 169 |
| 189 | |
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Common terms and phrases
anti-doubling property Assume Barlow comparison principle constant Corollary defined Definition denote diagonal lower estimate diagonal upper estimate diffusion Dirichlet Einstein relation electric network elliptic Harnack inequality equation equivalent Faber-Krahn inequality Fe W1 finite set fractals Green functions Green kernel harmonic function heat kernel estimates holds implies integer isoperimetric inequalities Laplace operator Lemma Let us observe Let us recall lower bound Markov chains Markov property Martina Franca mean exit mean value inequality non-negative notation obtain PLE F Poincaré inequality Proof Let Proof of Theorem Proposition PSM V F PSMV random walks Remark resistance satisfies scaling function statement follows strongly recurrent sub-Gaussian super-level sets Theory two-sided Vicsek tree volume growth weighted graph XC XC