Harnack inequalities and sub-Gaussian estimates for random walks

Grigor’yan, Alexander; Telcs, András

doi:10.1007/s00208-002-0351-3

Cited by 84 publications

(126 citation statements)

References 25 publications

Supporting

Mentioning

124

Contrasting

Order By: Relevance

“…The following lemma is a slight generalization of Proposition 4.1 of [GT2] and is proved similarly; note that no geometric assumptions on (G, E) are needed.…”

Section: Lemma 41 Suppose (G E A) Satisfies P Hi(β) Then (G E mentioning

confidence: 92%

“…Many properties of random walks on graphs satisfying PHI(β) are already quite well known; in particular one has good estimates on the transition probabilities p n (x, y). The following is the main theorem of [GT2]. (EHI) and (R β ).…”

Section: Introductionmentioning

confidence: 94%

“…The following result is proved in [GT2] and [T1], but as it has a simple direct proof which does not use (EHI), we present it here.…”

Section: Lemma 42 Suppose (G E A) Satisfies (Ehi) Then There Eximentioning

confidence: 99%

See 2 more Smart Citations

Stability of parabolic Harnack inequalities

Barlow

Bass

2003

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let (G, E) be a graph with weights {axy} for which a parabolic Harnack inequality holds with space-time scaling exponent β ≥ 2. Suppose {a xy } is another set of weights that are comparable to {axy}. We prove that this parabolic Harnack inequality also holds for (G, E) with the weights {a xy }. We also give stable necessary and sufficient conditions for this parabolic Harnack inequality to hold.

show abstract

“…The following lemma is a slight generalization of Proposition 4.1 of [GT2] and is proved similarly; note that no geometric assumptions on (G, E) are needed.…”

Section: Lemma 41 Suppose (G E A) Satisfies P Hi(β) Then (G E mentioning

confidence: 92%

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

Stability of parabolic Harnack inequalities

Barlow

Bass

2003

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Heat kernel bounds like (2.4) have been the subject of a great deal of research, not only on fractals but also on manifolds and graphs (see [9,24,10,11] and the references therein). Here we satisfy ourselves with noting that they are known for many interesting examples, including various p.c.f.…”

Section: A Smooth Bump From the Heat Operatormentioning

confidence: 99%

Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

Rogers

Strichartz

Teplyaev

2008

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. The heat semigroup (probabilistic) method is applicable to a more general class of metric measure spaces with Laplacian, including certain infinitely ramified fractals; however the cutoff technique involves some loss in smoothness. From the analytic approach we establish a Borel theorem for p.c.f. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decomposed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting.

show abstract

“…For developments paralleling the ideas and results of Sect. 2 we refer the reader to [28,29,30,52,53,98] and the references therein.…”

Section: Graphs Of Bounded Degreementioning

confidence: 99%