Explicit Formulas for Heat Kernels on Diamond Fractals

Ruiz, Patricia Alonso

doi:10.1007/s00220-018-3221-x

Cited by 29 publications

(16 citation statements)

References 39 publications

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“…Among these examples, the most notable are the Sierpinski gasket in harmonic coordinates [31,49,50,56,63,71,85], fractal quantum graphs [5] and diamond fractals [2, and references therein]. In particular, on diamond fractals [2] provides explicit formulas for the heat kernel, which allow for many computations relevant to our paper. On the Sierpinski gasket [85,Proposition 4.14] shows how to make computations at the dense set of junction points.…”

Section: Bv Classmentioning

confidence: 99%

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Ruiz¹,

Baudoin²,

Chen³

et al. 2018

Preprint

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We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class B 1,1/2 (X) that was introduced in our previous paper. Assuming furthermore a strong Bakry-Émery curvature type condition, we prove that for p > 1, the Sobolev class W 1,p (X) can be identified with B p,1/2 (X). Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.

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Section: Bv Classmentioning

confidence: 99%

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Ruiz¹,

Baudoin²,

Chen³

et al. 2018

Preprint

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show abstract

“…We study symmetric regular Dirichlet forms [26,29] on the fractallike spaces F ∞ constructed in [18]. Our motivation primarily comes from applications in mathematical physics, see [1][2][3][4][5][6]25, and references therein]. In particular, [6] shows that explicit formulas for kernels of spectral operators, such as heat kernel and Schroödinger kernels, can be obtained for these types of fractal spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Our motivation primarily comes from applications in mathematical physics, see [1][2][3][4][5][6]25, and references therein]. In particular, [6] shows that explicit formulas for kernels of spectral operators, such as heat kernel and Schroödinger kernels, can be obtained for these types of fractal spaces. The main results of our paper, Theorems 5.1 and 5.2, deal with the spectrum and the spectral resoltion of the Laplaican on the Barlow-Evans type projective limit space.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis on Barlow and Evans’ projective limit fractals

Steinhurst

Teplyaev

2021

J. Spectr. Theory

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We develop the foundation of the spectral analysis on Barlow-Evans projective limit fractals, or vermiculated spaces, which corresponds to symmetric Markov processes on these spaces. For some new examples, such as the generalized Laakso spaces and a Spierpinski Pâte à Choux, one can develop a complete spectral theory, including the eigenfunction expansions that are analogous to Fourier series. Also, one can construct connected fractal spaces isospectral to the fractal strings of Lapidus and van Frankenhuijsen. Our work is motivated by recent progress in mathematical physics on fractals.

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“…[21, Sec. 4] or [1]): one reason is that the approximating sequence of graphs has no uniform bound on the vertex degree. We partly treat such examples in [44].…”

Section: Introductionmentioning

confidence: 99%

Graph‐like spaces approximated by discrete graphs and applications

Post

Simmer

2021

Mathematische Nachrichten

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We define a distance between energy forms on a graph‐like metric measure space and on a suitable discrete weighted graph using the concept of quasi‐unitary equivalence. We apply this result to metric graphs, graph‐like manifolds (e.g. a small neighbourhood of an embedded metric graph) or pcf self‐similar fractals as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary), and express the distance of the associated energy forms in terms of geometric quantities. In particular, we show that there is a sequence of domains converging to a pcf self‐similar fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. As a consequence, the spectra and suitable functions of the associated Laplacians converge, the latter in operator norm.

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Explicit Formulas for Heat Kernels on Diamond Fractals

Cited by 29 publications

References 39 publications

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Spectral analysis on Barlow and Evans’ projective limit fractals

Graph‐like spaces approximated by discrete graphs and applications

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