Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs

  • 34 Pages
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by
Kyōto Daigaku Sūri Kaiseki Kenkyūjo , Kyoto, Japan
Statementby Martin T. Barlow, Thierry Coulhon and Takashi Kumagai.
SeriesRIMS -- 1467
ContributionsCoulhon, T., Kyōto Daigaku. Sūri Kaiseki Kenkyūjo.
Classifications
LC ClassificationsMLCSJ 2008/00102 (Q)
The Physical Object
Pagination34 p. ;
ID Numbers
Open LibraryOL16642620M
LC Control Number2008554240

Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs Martin T. Barlow ∗ Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, Canada [email protected] Thierry Coulhon † D´epartement de Math´ematiques, Universit´e de Cergy-Pontoise, Pontoise, France [email protected] Characterization of sub-Gaussianheat kernel estimates on strongly recurrent graphs Martin T.

Barlow1 University of British Columbia, Vancouver, V6T 1Z2, Canada SUB-GAUSSIAN HEAT KERNELS ON STRONGLY RECURRENT GRAPHS 3 Gaussian heat kernel upper and lower estimates to hold: hn C x 6 y D)M C V C x 6 n1 b. Sub-Gaussian estimates for random walks are typical of fractal graphs.

We char-acterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. 5 c Wiley Periodicals, Inc.

1 Introduction Statement of the main result Let G be an innite locally nite connected graph. It was proved in [11] that sub-Gaussian heat kernel estimates are equivalent to resistance estimates for random walks on fractal graph under strongly recurrent condition.

Hence we obtain sub. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Sub-Gaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities.

Details Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs PDF

Characterization of sub-Gaussian heat kernel By Martin T. Barlow, Thierry Coulhon and Takashi Kumagai. Abstract. Sub-Gaussian estimates for random walks are typical of fractal graphs.

We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs ABSTRACT: Sub-Gaussian estimates for random walks are typical of fractal graphs.

We characterize them in the strongly recurrent case, in terms of resistance estimates only. the present time, no similar characterization of the spaces with sub-Gaussian estimates seems to be known. All examples of spaces where () is proved are fractal-like spaces featuring a self-similarity structure.

The purpose of this paper is to provide a new approach to obtaining sub-Gaussian estimates of the heat kernel. Abstract Sub-Gaussian heat kernel estimates are typical of fractal graphs. We show that sub-Gaussian estimates on graphs follow from a Poincare inequality, capacity upper bound, and a slow volume growth condition.

An important feature of this work is that we do not assume elliptic Harnack inequality, cuto Sobolev inequality, or exit time bounds. The heart of the book is then devoted to the study of the heat kernel (Chapters 4, 5 and 6).

The author develops sufficient conditions under which sub-Gaussian or Gaussian bounds for the heat kernel hold (both on-diagonal and off diagonal; both upper and lower. M.T. Barlow, T. Coulhon, T. Kumagai, Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs.

Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs of sub-Gaussian heat kernel estimates on strongly recurrent graphs, Comm. Pure Appl. Math. LVIII ( Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs.

Description Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs PDF

Comm. Pure Appl. Math., 58 (), no. 12, PDF File (kb), Post Script File. This book has been cited by the following publications. graph directed fractals. Mathematische Annalen, Vol.Issue. 4, p. CrossRef; Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs.

Communications on Pure and Applied Mathematics, Vol. 58, Issue. 12, p. Grigor’yan, A., Kumagai, T.: On the dichotomy in the heat kernel two sided estimates. In: P. et al. (ed.) Analysis on Graphs and its Applications, Procedings of the Symposia in Pure Mathematics, American Mathematical Society, vol.

77, pp. – () Google Scholar. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs Barlow, M.

; Coulhon, Thierry ; Kumagai, T. The expanding dusty bipolar nebula around the. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Barlow, T.

Coulhon, T. Kumagai. Parabolic Harnack inequality and estimates of Markov chains on graphs. Delmotte. Harnack inequalities and sub-Gaussian estimates for random walks. Grigor'yan, A. Telcs. MR [BCK] Martin T.

Barlow, Thierry Coulhon, and Takashi Kumagai, Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs, Comm. Pure Appl. Math. 58 (), no. 12, – MRCiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

4. Very strongly recurrent graphs. Definition Following we say that a graph is very strongly recurrent (VSR) if there is a c > 0 such that for all x ∈ Γ, r > 0, w ∈ ∂ B (x, r) P w (τ x heat kernel lower bound for very strongly recurrent graphs.

[16] B. Hambly and T. Kumagai, Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries. Proc. of Symposia in Pure Math. 72, Part 2, pp. –, Amer. Math. Soc. As an application, we obtain sub-Gaussian heat kernel estimates of the time changed Brownian motions with respect to these measures.

The walk dimensions obtained under these new metrics are strictly greater than 2 and are closely related to the spectral dimension of fractal Laplacians. Alexander Grigor′yan and Andras Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math.

(), no. 3, – MR/S; Alexander Grigor’yan and András Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math.

Ann. (), no. 3, – We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large N the disconnection time of G N ×ℤ has rough order |G N | 2, when G N =(ℤ/Nℤ) agreement with a conjecture by I.

Benjamini, we show here that this behavior has broad generality when the. SUB-GAUSSIAN ESTIMATES OF HEAT KERNELS As we see, for groups of polynomial growth and for nonnegatively curved mani-folds, the heat kernel is fully determined (up to constant factors) by the volume growth function.

In other words, the potential theory on such spaces is characterized by a sin-gle parameter α—the exponent of the volume growth. Coulhon, Thierry. Coulhon, T. Coulhon, Thierry, Thierry Coulhon VIAF ID: (Personal) Permalink:   The fundamental solution of the heat equation (or heat kernel) on R n is given by the Gauss Weierstrass kernel p t (x, y) = 1 (4 π t) n / 2 exp ⁡ (− d (x, y) 2 4 t), for all x, y ∈ R n, t > 0.

From a probabilistic viewpoint, the heat kernel can be interpreted as the transition probability density of the diffusion generated by the. [81] Grigor’yan A., Telcs A., Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math.

J., () [82] Grigor’yan A., Telcs A., Harnack inequalities and sub-Gaussian estimates. Clone via HTTPS Clone with Git or checkout with SVN using the repository’s web address.

In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree n with an independent copy of a graph Gn and gluing the inserted graphs along the tree structure. We assume that there exist constants d,R ≥ 1, v.

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Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs Barlow, M.T.; Coulhon, T.; Kumagai, T. Wiley Series .normalization constant this Gaussian kernel is a normalized kernel, i.e. its integral over its full domain is unity for every s.

This means that increasing the s of the kernel reduces the amplitude substantially. Let us look at the graphs of the normalized kernels for s=s= 1 and s= 2 plotted on the same axes: [email protected] gaussD ;[email protected] x.In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical heat kernel represents the evolution of.