Volume Doubling Measures And Heat Kernel Estimates On Self Similar Sets

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Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets

Jun Kigami
Volume Doubling Measures and Heat Kernel Estimates on Self Similar Sets Book PDF

Author : Jun Kigami
Publisher : American Mathematical Soc.
Page : 110 pages
File Size : 41,6 Mb
Release : 2009-04-10
Category : Mathematics
ISBN : 9780821842928

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Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets by Jun Kigami Pdf

This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? These three problems turn out to be closely related. The author introduces a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and gives complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel.

Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates

Jun Kigami
Resistance Forms Quasisymmetric Maps and Heat Kernel Estimates Book PDF

Author : Jun Kigami
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 41,5 Mb
Release : 2012-02-22
Category : Mathematics
ISBN : 9780821852996

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Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates by Jun Kigami Pdf

Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow ``intrinsic'' with respect to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric. In this paper, the author considers the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. The author's main concerns are the following two problems: (I) When and how to find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes. (II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes.

Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance

Jun Kigami
Time Changes of the Brownian Motion Poincar Inequality Heat Kernel Estimate and Protodistance Book PDF

Author : Jun Kigami
Publisher : American Mathematical Soc.
Page : 118 pages
File Size : 43,9 Mb
Release : 2019-06-10
Category : Brownian motion processes
ISBN : 9781470436209

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Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance by Jun Kigami Pdf

In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n-dimensional cube [0,1]n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [0,1]n, density of the medium is homogeneous and represented by the Lebesgue measure. The author's study includes densities which are singular to the homogeneous one. He establishes a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [0,1]2 and self-similar measures. The author shows the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, he obtains diagonal lower and upper estimates of the heat kernel as time tends to 0. In particular, to express the principal part of the lower diagonal heat kernel estimate, he introduces “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen
Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II Book PDF

Author : David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 50,7 Mb
Release : 2013-10-24
Category : Mathematics
ISBN : 9780821891483

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Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II by David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen Pdf

This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoît Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry and various aspects of dynamical systems in applied mathematics and the applications to other sciences. Also included are articles discussing a variety of connections between these subjects and various areas of physics, engineering, computer science, technology, economics and finance, as well as of mathematics (including probability theory in relation with statistical physics and heat kernel estimates, geometric measure theory, partial differential equations in relation with condensed matter physics, global analysis on non-smooth spaces, the theory of billiards, harmonic analysis and spectral geometry). The companion volume (Contemporary Mathematics, Volume 600) focuses on the more mathematical aspects of fractal geometry and dynamical systems.

Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs

Alexander Grigor'yan,Yuhua Sun
Analysis and Partial Differential Equations on Manifolds Fractals and Graphs Book PDF

Author : Alexander Grigor'yan,Yuhua Sun
Publisher : Walter de Gruyter GmbH & Co KG
Page : 526 pages
File Size : 43,7 Mb
Release : 2021-01-18
Category : Mathematics
ISBN : 9783110700763

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Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs by Alexander Grigor'yan,Yuhua Sun Pdf

The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.

Geometry and Analysis of Metric Spaces via Weighted Partitions

Jun Kigami
Geometry and Analysis of Metric Spaces via Weighted Partitions Book PDF

Author : Jun Kigami
Publisher : Springer Nature
Page : 164 pages
File Size : 52,8 Mb
Release : 2020-11-16
Category : Mathematics
ISBN : 9783030541545

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Geometry and Analysis of Metric Spaces via Weighted Partitions by Jun Kigami Pdf

The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics

David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen
Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics Fractals in pure mathematics Book PDF

Author : David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 41,5 Mb
Release : 2013-10-22
Category : Mathematics
ISBN : 9780821891476

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Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics by David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen Pdf

This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoit Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry (and some aspects of dynamical systems) in pure mathematics. Also included are articles discussing a variety of connections of fractal geometry with other fields of mathematics, including probability theory, number theory, geometric measure theory, partial differential equations, global analysis on non-smooth spaces, harmonic analysis and spectral geometry. The companion volume (Contemporary Mathematics, Volume 601) focuses on applications of fractal geometry and dynamical systems to other sciences, including physics, engineering, computer science, economics, and finance.

Stability of Heat Kernel Estimates for Symmetric Non-Local Dirichlet Forms

Zhen-Qing Chen,Takashi Kumagai,Jian Wang
Stability of Heat Kernel Estimates for Symmetric Non Local Dirichlet Forms Book PDF

Author : Zhen-Qing Chen,Takashi Kumagai,Jian Wang
Publisher : American Mathematical Society
Page : 89 pages
File Size : 44,8 Mb
Release : 2021-09-24
Category : Mathematics
ISBN : 9781470448639

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Stability of Heat Kernel Estimates for Symmetric Non-Local Dirichlet Forms by Zhen-Qing Chen,Takashi Kumagai,Jian Wang Pdf

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Quantum Graphs and Their Applications

Gregory Berkolaiko,Robert Carlson,Stephen A. Fulling
Quantum Graphs and Their Applications Book PDF

Author : Gregory Berkolaiko,Robert Carlson,Stephen A. Fulling
Publisher : American Mathematical Soc.
Page : 322 pages
File Size : 54,6 Mb
Release : 2006
Category : Quantum graphs
ISBN : 9780821837658

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Quantum Graphs and Their Applications by Gregory Berkolaiko,Robert Carlson,Stephen A. Fulling Pdf

This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of mathematics: number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.

Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

Volkmar Liebscher
Random Sets and Invariants for Type II Continuous Tensor Product Systems of Hilbert Spaces Book PDF

Author : Volkmar Liebscher
Publisher : American Mathematical Soc.
Page : 124 pages
File Size : 51,6 Mb
Release : 2009-04-10
Category : Mathematics
ISBN : 9780821843185

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Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces by Volkmar Liebscher Pdf

In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in $[0,1]$ or $\mathbb R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types $\mathrm{I}_n$, $\mathrm{II}_n$ and $\mathrm{III}$ is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.

Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three

Robert C. Dalang,Marta Sanz SolŽ
Holder Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three Book PDF

Author : Robert C. Dalang,Marta Sanz SolŽ
Publisher : American Mathematical Soc.
Page : 83 pages
File Size : 48,6 Mb
Release : 2009-04-10
Category : Mathematics
ISBN : 9780821842881

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Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three by Robert C. Dalang,Marta Sanz SolŽ Pdf

The authors study the sample path regularity of the solution of a stochastic wave equation in spatial dimension $d=3$. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. The authors prove that at any fixed time, a.s., the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any fixed $x\in\mathbb{R}^3$, the sample paths in time are Holder continuous functions. Further, the authors obtain joint Holder continuity in the time and space variables. Their results rely on a detailed analysis of properties of the stochastic integral used in the rigourous formulation of the s.p.d.e., as introduced by Dalang and Mueller (2003). Sharp results on one- and two-dimensional space and time increments of generalized Riesz potentials are a crucial ingredient in the analysis of the problem. For spatial covariances given by Riesz kernels, the authors show that the Holder exponents that they obtain are optimal.

Heat Eisenstein Series on $\mathrm {SL}_n(C)$

Jay Jorgenson,Serge Lang
Heat Eisenstein Series on mathrm SL n C Book PDF

Author : Jay Jorgenson,Serge Lang
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 49,5 Mb
Release : 2009
Category : Decomposition
ISBN : 9780821840443

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Heat Eisenstein Series on $\mathrm {SL}_n(C)$ by Jay Jorgenson,Serge Lang Pdf

The purpose of this Memoir is to define and study multi-variable Eisenstein series attached to heat kernels. Fundamental properties of heat Eisenstein series are proved, and conjectural behavior, including their role in spectral expansions, are stated.

Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules

AndrŽ Martinez,Vania Sordoni
Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules Book PDF

Author : AndrŽ Martinez,Vania Sordoni
Publisher : American Mathematical Soc.
Page : 96 pages
File Size : 54,6 Mb
Release : 2009-06-05
Category : Mathematics
ISBN : 9780821842966

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Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules by AndrŽ Martinez,Vania Sordoni Pdf

The authors construct an abstract pseudodifferential calculus with operator-valued symbol, suitable for the treatment of Coulomb-type interactions, and they apply it to the study of the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case of the electronic Hamiltonian admitting a local gap in its spectrum. In particular, they show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, they study the propagation of certain wave packets up to long time values of Ehrenfest order.

Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves

GŽrard Iooss,Pavel I. Plotnikov
Small Divisor Problem in the Theory of Three Dimensional Water Gravity Waves Book PDF

Author : GŽrard Iooss,Pavel I. Plotnikov
Publisher : American Mathematical Soc.
Page : 144 pages
File Size : 53,5 Mb
Release : 2009-06-05
Category : Science
ISBN : 9780821843826

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Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves by GŽrard Iooss,Pavel I. Plotnikov Pdf

The authors consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta$ between them. Denoting by $\mu =gL/c^{2}$ the dimensionless bifurcation parameter ( $L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), bifurcation occurs for $\mu = \cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. ``Diamond waves'' are a particular case of such waves, when they are symmetric with respect to the direction of propagation. The main object of the paper is the proof of existence of such symmetric waves having the above mentioned asymptotic expansion. Due to the occurence of small divisors, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta$, the 3-dimensional travelling waves bifurcate for a set of ``good'' values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane $(\theta,\mu ).$