Brownian Motion and Harmonic Analysis on Sierpinski Carpets
نویسندگان
چکیده
منابع مشابه
Brownian Motion and Harmonic Analysis on Sierpinski Carpets
We consider a class of fractal subsets of Rd formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and...
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We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined. Research partially supported by NSERC (Canada), and EPSRC (UK). Research partially...
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We consider random walks on a class of graphs derived from Sierpinski carpets. We obtain upper and lower bounds (which are non-Gaussian) on the transition probabilities which are, up to constants, the best possible. We also extend some classical Sobolev and Poincar e inequalities to this setting.
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The Lie group Sol(p, q) is the semidirect product induced by the action of R on R which is given by (x, y) 7→ (ex, e−qzy), z ∈ R. Viewing Sol(p, q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and...
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Uniform Harnack inequalities for harmonic functions on the preand graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in Md , d > 3, estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities. The Sierpinski carpets (S...
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ژورنال
عنوان ژورنال: Canadian Journal of Mathematics
سال: 1999
ISSN: 0008-414X,1496-4279
DOI: 10.4153/cjm-1999-031-4