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 selfsimilarity. The author studies the heat diffusion at this general level and discusses the following topics:

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antidoubling property apply Assume Barlow Cetraro Cexp comparison principle constant Corollary deﬁned Deﬁnition denote diagonal lower estimate diagonal upper estimate diﬀusion Dirichlet Einstein relation electric network elliptic Harnack inequality equation FaberKrahn inequality ﬁnite finite sets ﬁrst fractals function F Green functions Green kernel harmonic function heat kernel estimates implies inﬁnite integer isoperimetric inequalities iterating Laplace operator Lemma Let us observe Let us recall lower bound Markov chains Markov property Martina Franca mean exit mean value inequality nonnegative notation obtain potential potential theory Proof Let Proof of Theorem Proof The proof Proof The statement Proposition proved PSMV F random walks Remark resistance satisfies scaling function Sobolev inequality statement follows strongly recurrent subGaussian superlevel sets Theory twosided upper bound Vicsek tree volume growth weighted graph whence yeB(x,R