Dirichlet Forms and Related Topics

Name: Dirichlet Forms and Related Topics | In Honor of Masatoshi Fukushima's Beiju, IWDFRT 2022, Osaka, Japan, August 22-26
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In Honor of Masatoshi Fukushima's Beiju, IWDFRT 2022, Osaka, Japan, August 22-26

Zhen-Qing Chen Masayoshi Takeda Toshihiro Uemura(Editor)

Springer (Publisher)

1st Edition

Published on 4. September 2022

XXI, 572 pages

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978-981-19-4672-1 (ISBN)

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Intro
Preface
List of Masatoshi Fukushima's Publications
Contents
Markov Uniqueness and Fokker-Planck-Kolmogorov Equations
1 Introduction and Framework
2 The Main Idea and a Parabolic Condition for Uniqueness
3 Some Uniqueness Results for FPKEs
3.1 Fokker-Planck-Kolmogorov Equations
3.2 Nondegenerate VMO Diffusion Coefficients
3.3 Nondegenerate Locally Lipschitz Diffusion Coefficients
3.4 Nondegenerate Diffusion Coefficients and the Lyapunov Function Condition
3.5 Degenerate Diffusion Coefficients
4 Applications to the Markov Uniqueness Problem
4.1 The Framework
4.2 Nondegenerate VMO Diffusion Coefficients
4.3 Nondegenerate Locally Lipschitz Diffusion Coefficients
4.4 Nondegenerate Diffusion Coefficients and Lyapunov Function Conditions
4.5 Degenerate Diffusion Coefficients
References
A Chip-Firing and a Riemann-Roch Theorem on an Ultrametric Space
1 Introduction
2 Laplacian and Riemann-Roch Theorem on Ultrametric Space with Finite Vertices
3 Unification of Ultrametric Space with Finite Verteces
4 Riemann-Roch Theorem on Ultrametric Space with Countably Many Vertices
References
Hermitizable, Isospectral Matrices or Differential Operators
1 Hermitizable, Isospectral Matrices
2 Hermitizable, Isospectral Differential Operators
References
On Strongly Continuous Markovian Semigroups
References
Two-Sided Heat Kernel Estimates for Symmetric Diffusion Processes with Jumps: Recent Results
1 Introduction
2 Stability of Heat Kernel Estimates for Symmetric Diffusions with Jumps
2.1 Two-Sided Heat Kernel Estimates
2.2 Example
3 Symmetric Reflected Diffusions with Jumps in Inner Uniform Domains
3.1 Reflected Diffusions on Inner Uniform Domains
3.2 Reflected Diffusions with Jumps
3.3 Heat Kernel Estimates for the beta Subscript asterisk Baseline less than or equals beta Superscript asterisk Baseline less than or equals normal infinityß*leß* leinfty in StartSet 0 Subscript plus Baseline EndSet union left parenthesis 0 comma normal infinity right bracket{0+}(0, infty] Case
3.4 Discussion on Off-Diagonal Heat Kernel Upper Bound
3.5 Example
References
On Non-negative Solutions to Space-Time Partial Differential Equations of Higher Order
1 Introduction
2 An Abstract Problem
3 Some Translation Invariant Pseudo-differential Operators
4 Some Discussions on the Case upper N equals 2N=2
5 Higher Order Partial Differential Equations Admitting Non-negative Solutions
References
Monotonicity Properties of Regenerative Sets and Lorden's Inequality
1 Introduction
2 Lorden's Inequality
3 Monotone Potential Density
4 Concluding Remarks
References
Doob Decomposition, Dirichlet Processes, and Entropies on Wiener Space
1 Introduction
2 Doob Decomposition in Continuous Time
3 Entropies and Couplings on Wiener Space
References
Analysis on Fractal Spaces and Heat Kernels
1 Introduction
2 Classical Heat Kernel
3 Examples of Fractals
4 Dirichlet Forms
5 Walk Dimension
6 Besov Spaces and Characterization of betaß
7 Dichotomy of Self-similar Heat Kernels
8 Estimating Heat Kernels: Strongly Local Case
9 Estimating Heat Kernels: Jump Case
10 Ultra-metric Spaces
References
Silverstein Extension and Fukushima Extension
1 Introduction
2 Silverstein Extensions
3 Fukushima Extensions
4 Examples of Fukushima Extensions
5 Fukushima Subspaces
6 Example: Revisit
References
Singularity of Energy Measures on a Class of Inhomogeneous Sierpinski Gaskets
1 Introduction
2 Framework and Statement of Theorems
3 Preliminary Lemmas
4 Proof of the Main Results
5 Concluding Remarks
References
On upper L Superscript pLp Liouville Theorems for Dirichlet Forms
1 Introduction
2 Fukushima's Ergodic Theorem
3 Regular Dirichlet Forms and Harmonic Functions
3.1 Basic Notions and Intrinsic Metrics
3.2 Harmonic Functions
4 A Caccioppoli Inequality
5 Proof of Yau's and Karp's Theorem and Recurrence
5.1 Proof of Yau's and Karp's Theorem
5.2 Proof of the Growth Test for Recurrence
References
On Singularity of Energy Measures for Symmetric Diffusions with Full Off-Diagonal Heat Kernel Estimates II: Some Borderline Examples
1 Introduction
2 The Examples: Thin Scale Irregular Sierpinski Gaskets
3 Space-Time Scale Function upper Psi Subscript bold italic ll and fHKEfHKE left parenthesis upper Psi Subscript bold italic l Baseline right parenthesisfHKE(l)
4 Singularity of the Energy Measures
5 Realizing Arbitrarily Slow Decay Rates of upper Psi left parenthesis r right parenthesis divided by r squared(r)/r2
References
Scattering Lengths for Additive Functionals and Their Semi-classical Asymptotics
1 Introduction
2 Scattering Length for Additive Functionals
3 Kac's Scattering Length Formula
4 Semi-classical Asymptotics for Scattering Length
References
Equivalence of the Strong Feller Properties of Analytic Semigroups and Associated Resolvents
1 Introduction
2 Preliminaries
3 Equivalence of the Strong Feller Properties
4 Application to Markov Processes Associated with Lower Bounded Semi-Dirichlet Forms
References
Interactions Between Trees and Loops, and Their Representation in Fock Space
1 Framework and Definitions
2 Interaction Between Tree and Loops
3 Fock Spaces
4 Local Interaction in Supersymmetric Fock Space
References
Remarks on Quasi-regular Dirichlet Subspaces
1 Introduction
2 Quasi-regular Dirichlet Subspaces
3 Quasi-regular Dirichlet Subspaces of Concrete Dirichlet Forms
3.1 One-Dimensional Brownian Motion
3.2 Multi-dimensional Brownian Motion
4 Further Remarks
References
Power-Law Dynamic Arising from Machine Learning
1 Introduction
2 Background and Preliminaries on Power-Law Dynamic
2.1 Background in Machine Learning
2.2 Preliminaries on Power-Law Dynamic
3 Property of the Stationary Distribution
4 Existence and Uniqueness of the Stationary Distribution
5 First Exit Time: Asymptotic Order
6 First Exit Time: From Continuous to Discrete
References
Hölder Estimates for Resolvents of Time-Changed Brownian Motions
1 Introduction
2 Main Results
3 Preliminary Lemmas
4 Proof of Theorem 1
References
On the Continuity of Half-Plane Capacity with Respect to Carathéodory Convergence
1 Introduction
2 Study on the Upper Half-Plane
2.1 Basic Definitions and Proof of Theorem1
2.2 Strict Monotonicity
3 Study on Parallel Slit Half-Planes
3.1 BMD Half-Plane Capacity
3.2 Markov Chains Induced by BMD
3.3 Uniform Regularity of Slit Domains
4 Relation to Geometric Function Theory
4.1 Half-Plane Capacity and Angular Residue at Infinity
4.2 Carathéodory Convergence and Locally Uniform Convergence
References
Dyson's Model in Infinite Dimensions Is Irreducible
1 Introduction
2 The MathID170 m-Labeled Process and the Lyons-Zheng Decomposition
3 Proof of Theorems 2 and 3
4 Proof of Theorem 1
References
(Weak) Hardy and Poincaré Inequalities and Criticality Theory
1 Introduction
2 Preliminaries
2.1 Closed Quadratic Forms on upper L squared left parenthesis upper X comma mu right parenthesisL2(X,µ) and upper L Superscript 0 Baseline left parenthesis upper X comma mu right parenthesisL0(X,µ)
2.2 Extensions of Positivity Preserving Operators
3 The Beurling-Deny Criteria, Excessive Functions and Extended Forms
3.1 Basics and Excessive Functions
3.2 The Extensions q Subscript eqe and q Superscript plusq+
3.3 Invariant Sets and Irreducibilty
4 (Very) Weak and Abstract Poincaré and Hardy inequalities
5 From Weak Hardy Inequalities to Hardy Inequalities-Subcriticality
6 Weak Poincaré Inequalities and Completeness of Extended form Domains
References
Maximal Displacement of Branching Symmetric Stable Processes
1 Introduction
2 Symmetric Stable Processes
2.1 Resolvent Asymptotics
2.2 Spectral Properties of Schrödinger Type Operators with the Fractional Laplacian
2.3 Asymptotic Behaviors of Feynman-Kac Functionals
3 Maximal Displacement of Branching Symmetric Stable Processes
3.1 Branching Symmetric Stable Processes
3.2 Weak Convergence and Tail Asymptotics
3.3 Examples
4 Proof of Theorem 17
References
Random Riemannian Geometry in 4 Dimensions
1 Random Riemannian Geometries and Conformal Invariance
2 Paneitz Energy on 4-Dimensional Manifolds
3 Co-biharmonic Gaussian Field and Quantum Liouville Measure
3.1 Conformally Invariant Gaussian Field
3.2 Quantum Liouville Measure
4 Approximation by Random Fields and Liouville Measures on the Discrete 4-Torus
4.1 The Isotropic Haar System
4.2 The Semi-discrete Gaussian Field
4.3 The Semi-discrete Liouville Measure
4.4 Discrete Random Objects
References
Infinite Particle Systems with Hard-Core and Long-Range Interaction
1 Introduction
2 Preliminaries
2.1 Systems of Unlabeled Hard Balls
2.2 Systems of Labeled Balls
2.3 Skorohod Equation
3 Results
3.1 Existence of a Weak Solution
3.2 Statement of the Results
4 Proof of the Main Theorem
4.1 Finite Cluster Property
4.2 On the Lipschitz Continuity of b Subscript bold italic upper X Superscript double struck upper IbXmathbbI
4.3 German upper ImathfrakI-IFC
4.4 Proof of Theorem3.3
References
On Universality in Penalisation Problems with Multiplicative Weights
1 Introduction
2 Penalised Probability
3 Subsequent Markov Property
4 Universality Class
5 Penalisation Problems
5.1 Constant Clock
5.2 Exponential Clock
6 Brownian Penalisation Revisited
7 Lévy Penalisation Revisited
8 Langevin Penalisation Revisited
References
Asymptotic Behavior of Spectral Functions for Schrödinger Forms with Signed Measures
1 Introduction
2 Preliminaries
3 The Spectral Function of a Schrödinger Form with Positive Measures
4 The Spectral Function of a Schrödinger Form with a Signed Measure
References

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Dirichlet Forms and Related Topics

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