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Naotaka Kajino (2020)
An elementary proof of walk dimension being greater than two for Brownian motion on Sierpi\'{n}ski carpetsarXiv: Probability
Rajeev Atla, Quentin Dubroff (1987)
Random walks and electric networksAmerican Mathematical Monthly, 94
M. Barlow, R. Bass, T. Kumagai, A. Teplyaev (2008)
Uniqueness of Brownian motion on Sierpinski carpetsJournal of the European Mathematical Society, 12
Eric Mbakop (2009)
Analysis on Fractals
Denali Molitor, Nadia Ott, R. Strichartz (2014)
USING PEANO CURVES TO CONSTRUCT LAPLACIANS ON FRACTALSFractals, 23
M. Barlow, R. Bass (1992)
Transition densities for Brownian motion on the Sierpinski carpetProbability Theory and Related Fields, 91
Tyrus Berry, Steven Heilman, R. Strichartz (2009)
Outer Approximation of the Spectrum of a Fractal LaplacianExperimental Mathematics, 18
K Andrews, A. Ulysses (2017)
Existence of Diffusions on 4N Carpets
M. Barlow (1998)
Diffusions on fractals
E. Stein (1971)
Singular Integrals and Di?erentiability Properties of Functions
I. McGillivray (2002)
Resistance in Higher-Dimensional Sierpiński CarpetsPotential Analysis, 16
A. Jonsson, H. Wallin, J. Peetre (1984)
Function spaces on subsets of Rn
Kalamazoo College Email address: orwinmc@gmail
M. Barlow, R. Bass (1989)
The construction of brownian motion on the Sierpinski carpetAnnales De L Institut Henri Poincare-probabilites Et Statistiques, 25
D. Kelleher, A. Brzoska, Hugo Panzo, A. Teplyaev (2015)
Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisionsDiscrete & Continuous Dynamical Systems - S
Naotaka Kajino (2010)
Spectral asymptotics for Laplacians on self-similar setsJournal of Functional Analysis, 258
M. Barlow, R. Bass (1999)
Brownian Motion and Harmonic Analysis on Sierpinski CarpetsCanadian Journal of Mathematics, 51
J. Cooper (1973)
SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONSBulletin of The London Mathematical Society, 5
M. Barlow, R. Bass (1990)
On the resistance of the Sierpiński carpetProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 431
(1989)
Sobolev spaces and functions of bounded variation
(1989)
Weakly differentiable functions , volume 120 of Graduate Texts in Mathematics
Rochester Institute of Technology Email address: clairemcanner@gmail
A. Grigor’yan, Meng Yang (2017)
Local and non-local Dirichlet forms on the Sierpiński carpetTransactions of the American Mathematical Society
P. Grisvard (1985)
Elliptic Problems in Nonsmooth Domains
(1981)
Fractals and self-similarity
L. Evans (1992)
Measure theory and fine properties of functions
W. Ziemer (1989)
Weakly differentiable functions
R. Brown (1994)
The mixed problem for laplace's equation in a class of lipschitz domainsCommunications in Partial Differential Equations, 19
M. Barlow, R. Bass, J. Sherwood (1990)
Resistance and spectral dimension of Sierpinski carpetsJournal of Physics A, 23
E. Stein (1971)
Singular Integrals and Differentiability Properties of Functions (PMS-30), Volume 30
AbstractThe 4N{4N}-carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4N{4N}-carpet F, let {Fn}n≥0{\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂nFn=F{\bigcap_{n}F_{n}=F}. On each Fn{F_{n}}, let ℰ(u,v)=∫FN∇u⋅∇vdx{\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and un{u_{n}} be the unique harmonic function on Fn{F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet,Proc. Roy. Soc. Lond. Ser. A431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ=ρ(N)>1{\rho=\rho(N)>1} such that ℰ(un,un)ρn{\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.
Forum Mathematicum – de Gruyter
Published: Jan 1, 2022
Keywords: Resistance; fractal; fractal carpet; Dirichlet form; walk dimension; spectral dimension; 28A80; 31C25; 31E05; 31C15; 60J65
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