Dirichlet forms on fractals: Poincaré constant and resistance

Kusuoka, Shigeo; Yin, Zhou Xian

doi:10.1007/bf01195228

Cited by 140 publications

(207 citation statements)

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“…The latter means that the process X t , started at a point x, first exits the ball B(x, R) at the time t R 2 . The development of Markov processes on fractals and fractal-like graphs (see [10], [7], [30], [36], [40], [41], [42], [44], [45], [59], [67], etc.) has led to construction of homogeneous metric spaces M where the process X t has the diffusion speed of the order t 1/β , with some β > 2.…”

Section: Introduction Consider the Heat Equationmentioning

confidence: 99%

Sub-Gaussian estimates of heat kernels on infinite graphs

Grigor’yan

Telcs²

2001

Duke Math. J.

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Section: Introduction Consider the Heat Equationmentioning

confidence: 99%

Sub-Gaussian estimates of heat kernels on infinite graphs

Grigor’yan

Telcs²

2001

Duke Math. J.

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“…Sierpiński carpets and graphical Sierpiński carpets. These are examples of non-finitely-ramified fractals and fractal graphs [3,4,5,6,26]. In particular, they are non-p.c.f.…”

Section: Applications and Examplesmentioning

confidence: 99%

“…Consequently, Theorem 4 applies to all unbounded spaces constructed on GSC, while Theorem 6 applies to all Sierpiński carpets for which such estimates exist. Moreover, Theorem 1 applies to resistance Dirichlet forms on the Sierpiński carpets in dimension less than 2, such as self-similar Dirichlet forms on the Sierpiński carpets constructed in [26].…”

Section: Applications and Examplesmentioning

confidence: 99%

Weak uncertainty principle for fractals, graphs and metric measure spaces

Okoudjou

Saloff‐Coste

Teplyaev

2008

Trans. Amer. Math. Soc.

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Abstract. We develop a new approach to formulate and prove the weak uncertainty inequality, which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.

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“…Subsequently, Kusuoka and Zhou in [27] gave a different construction of a diffusion on F SC , which yielded a process that, as well as having the invariance properties of the Brownian motion constructed in [3], was also scale invariant. The proofs in [3,27] also work for fractals that are formed in a similar manner to the standard Sierpiński carpet: we call these generalized Sierpiński carpets (GSCs).…”

Section: Introductionmentioning

confidence: 99%

“…The proofs in [3,27] also work for fractals that are formed in a similar manner to the standard Sierpiński carpet: we call these generalized Sierpiński carpets (GSCs). In [5] the results of [3] were extended to GSCs embedded in R d for d ≥ 3.…”

Section: Introductionmentioning

confidence: 99%