Journal ArticleDOI
Transition densities for Brownian motion on the Sierpinski carpet
Martin T. Barlow,Richard F. Bass +1 more
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In this article, upper and lower bounds for the transition densitiesp(t, x, y) of Brownian motion on the Sierpinski carpet were obtained, and the existence of the spectral dimension was established.Abstract:
Upper and lower bounds are obtained for the transition densitiesp(t, x, y) of Brownian motion on the Sierpinski carpet. These are of the same form as those which hold for the Sierpinski gasket. In addition, the joint continuity ofp(t, x, y) is proved, the existence of the spectral dimension is established, and the Einstein relation, connecting the spectral dimension, the Hausdorff dimension and the resistance exponent, is shown to hold.read more
Citations
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Brownian Motion and Harmonic Analysis on Sierpinski Carpets
Martin T. Barlow,Richard F. Bass +1 more
TL;DR: In this paper, a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet is considered, and a uniform Harnack inequality for positive harmonic functions and a heat equation are derived.
Journal ArticleDOI
Non-local dirichlet forms and symmetric jump processes
TL;DR: In this paper, the authors considered the non-local symmetric Dirichlet form (E,F) given by with F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where E 1 (f, f):= E(f,f) + f Rd f(x) 2 dx, and the jump kernel J satisfies for 0 < α < β < 2, |x - y| < 1.
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Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals
Jun Kigami,Michel L. Lapidus +1 more
TL;DR: In this article, the authors established an analogue of Weyl's classical theorem for the asymptotics of eigenvalues of Laplacians on a finitely ramified (i.e., p.c.f.) self-similar fractal, such as, for example, the Sierpinski gasket.
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Random walks on supercritical percolation clusters
TL;DR: In this article, Gaussian upper and lower bounds on the transition density qt(x;y) of the continuous time simple random walk on a supercritical percolation cluster C1 in the Euclidean lattice were obtained.
References
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Journal ArticleDOI
Diffusion in disordered media
Shlomo Havlin,Daniel ben-Avraham +1 more
TL;DR: In this article, scaling theories and numerical simulations are used to describe diffusion processes on percolation systems and fractals, and different types of disordered systems exhibiting anomalous diffusion are presented.
Journal ArticleDOI
Random walks on fractal structures and percolation clusters
R. Rammal,Gérard Toulouse +1 more
TL;DR: The notion of spectral dimensionality of a self-similar (fractal) structure is recalled and its value for the family of Sierpinski gaskets derived via a scaling argument is derived in this paper.
Journal ArticleDOI
Brownian motion on the Sierpinski gasket
Martin T. Barlow,Edwin Perkins +1 more
TL;DR: In this paper, the Sierpinski gasket has been used to construct a Brownian motion, a diffusion process characterized by local isotropy and homogeneity properties, and it is shown that the process has a continuous symmetric transition density, p