Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999/08/01 by Martin T. Barlow, Richard F. Bass · 316 citations
Mathematics · Physics and Astronomy · #Brownian motion #Class (philosophy) #Fractal #Harmonic #Harmonic function #Harnack's inequality #Harnack's principle #Heat equation #Heat kernel #Isotropy #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Potential theory #Pure mathematics #Quantum chaos and dynamical systems #Sierpinski carpet #Sierpinski triangle #Statistics #Theoretical and Computational Physics

paper · pdf · doi:10.4153/cjm-1999-031-4

published in Canadian Journal of Mathematics 51(4), 673-744 (Cambridge University Press)

openalex publication_date 1999/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

Abstract We consider a class of fractal subsets of d 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 extend some classical Sobolev and Poincaré inequalities to this setting.

Citations

• Transition density estimates for Brownian motion on affine nested fractals
• A new proof of Moser's parabolic harnack inequality using the old ideas of Nash
• The Fractal Geometry of Nature
• Diffusion in disordered media
• Diffusion in disordered media

Cited by

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• Sobolev inequalities on product Sierpinski spaces
• L^p-Kato class measures for symmetric Markov processes under heat kernel estimates
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• A non-canonical diffusion on the Sierpiński carpet
• Fluctuations of recentered maxima of discrete Gaussian Free Fields on a class of recurrent graphs
• Exact mean first-passage time on generalized Vicsek fractal
• On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates
• Local and Non-Local Dirichlet Forms on the Sierpiński Gasket and the Sierpiński Carpet
• Boundary Harnack Principle for fractional powers of Laplacian on the Sierpinski carpet
• A guide to Brownian motion and related stochastic processes
• Discrete Approximations of Metric Measure Spaces of Controlled Geometry
• A weak type (p,a) criterion for operators, and applications
• Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces
• Bottom crossing probability for symmetric jump processes (full version)
• Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension
• A stability theorem for elliptic Harnack inequalities
• Outer Approximation of the Spectrum of a Fractal Laplacian
• Laws of the iterated logarithm for symmetric jump processes
• The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
• Dimension results and local times for superdiffusions on fractals
• Energy measure closability for Dirichlet forms
• Scale-invariant boundary Harnack principle on inner uniform domains in fractal-type spaces
• Reducible Iterated Graph Systems: multiscale-freeness and multifractals
• Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms
• Spectral Analysis and Dirichlet Forms on Barlow-Evans Fractals
• Resistance Scaling on 4N-Carpets
• Dirichlet forms on unconstrained Sierpinski carpets
• Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
• Weak uncertainty principle for fractals, graphs and metric measure spaces
• A Localized Diffusive Time Exponent for Compact Metric Spaces
• Mosco Convergence of Stable-Like Non-Local Dirichlet Forms on Metric Measure Spaces
• Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions
• The rate of decay of the Wiener sausage in local Dirichlet space
• The scaling limit of loop-erased random walk in three dimensions
• Upper estimate of martingale dimension for self-similar fractals

Related

• A Sierpinski carpet like fractal without standard self-similar energy
• A non-canonical diffusion on the Sierpiński carpet
• Some function spaces related to the Brownian motion on simple nested fractals
• Local and Non-Local Dirichlet Forms on the Sierpiński Gasket and the Sierpiński Carpet
• An elementary proof that walk dimension is greater than two for Brownian motion on Sierpiński carpets