Table Of ContentULTRAMETRIC PSEUDODIFFERENTIAL
EQUATIONS AND APPLICATIONS
Starting from physical motivations and leading to practical applications, this book
provides an interdisciplinary perspective on the cutting edge of ultrametric
pseudodifferential equations. It shows the ways in which these equations link
different ields, including mathematics, engineering, and geophysics. In particular,
the authors provide a detailed explanation of the geophysical applications of p-adic
diffusion equations useful when modeling the lows of liquids through porous rock.
p-Adic wavelets theory and p-adic pseudodifferential equations are also presented,
along with their connections to mathematical physics, representation theory, the
physics of disordered systems, probability, number theory, and p-adic dynamical
systems.
Material that was previously spread across many articles in journals of many
different ields is brought together here, including recent work on the van der Put
series technique. This book provides an excellent snapshot of the fascinating ield of
ultrametric pseudodifferential equations, including their emerging applications and
currently unsolved problems.
Encyclopedia of Mathematics and Its Applications
This series is devoted to signiicant topics or themes that have wide application in
mathematics or mathematical science and for which a detailed development of the
abstract theory is less important than a thorough and concrete exploration of the
implications and applications.
Books in the Encyclopedia of Mathematics and Its Applications cover their
subjects comprehensively. Less important results may be summarized as exercises
at the ends of chapters. For technicalities, readers can be referred to the
bibliography, which is expected to be comprehensive. As a result, volumes are
encyclopedic references or manageable guides to major subjects.
Encyclopedia of Mathematics and its Applications
All the titles listed below can be obtained from good booksellers or from Cambridge
University Press. For a complete series listing visit
www.cambridge.org/mathematics.
119 M. Deza and M. Dutour Sikiric´ Geometry of Chemical Graphs
120 T. Nishiura Absolute Measurable Spaces
121 M. Prest Purity, Spectra and Localisation
122 S. Khrushchev Orthogonal Polynomials and Continued Fractions
123 H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity
124 F. W. King Hilbert Transforms I
125 F. W. King Hilbert Transforms II
126 O. Calin and D.-C. Chang Sub-Riemannian Geometry
127 M. Grabisch et al. Aggregation Functions
128 L. W. Beineke and R. J. Wilson (eds.) with J. L. Gross and T. W. Tucker Topics in Topological Graph
Theory
129 J. Berstel, D. Perrin and C. Reutenauer Codes and Automata
130 T. G. Faticoni Modules over Endomorphism Rings
131 H. Morimoto Stochastic Control and Mathematical Modeling
132 G. Schmidt Relational Mathematics
133 P. Kornerup and D. W. Matula Finite Precision Number Systems and Arithmetic
134 Y. Crama and P. L. Hammer (eds.) Boolean Models and Methods in Mathematics, Computer Science, and
Engineering
135 V. Berthé and M. Rigo (eds.) Combinatorics, Automata and Number Theory
136 A. Kristály, V. D. Radulescu and C. Varga Variational Principles in Mathematical Physics, Geometry, and
Economics
137 J. Berstel and C. Reutenauer Noncommutative Rational Series with Applications
138 B. Courcelle and J. Engelfriet Graph Structure and Monadic Second-Order Logic
139 M. Fiedler Matrices and Graphs in Geometry
140 N. Vakil Real Analysis through Modern Ininitesimals
141 R. B. Paris Hadamard Expansions and Hyperasymptotic Evaluation
142 Y. Crama and P. L. Hammer Boolean Functions
143 A. Arapostathis, V. S. Borkar and M. K. Ghosh Ergodic Control of Diffusion Processes
144 N. Caspard, B. Leclerc and B. Monjardet Finite Ordered Sets
145 D. Z. Arov and H. Dym Bitangential Direct and Inverse Problems for Systems of Integral and Differential
Equations
146 G. Dassios Ellipsoidal Harmonics
147 L. W. Beineke and R. J. Wilson (eds.) with O. R. Oellermann Topics in Structural Graph Theory
148 L. Berlyand, A. G. Kolpakov and A. Novikov Introduction to the Network Approximation Method for
Materials Modeling
149 M. Baake and U. Grimm Aperiodic Order I: A Mathematical Invitation
150 J. Borwein et al. Lattice Sums Then and Now
151 R. Schneider Convex Bodies: The Brunn–Minkowski Theory (Second Edition)
152 G. Da Prato and J. Zabczyk Stochastic Equations in Ininite Dimensions (Second Edition)
153 D. Hofmann, G. J. Seal and W. Tholen (eds.) Monoidal Topology
154 M. Cabrera García and Á. Rodríguez Palacios Non-Associative Normed Algebras I: The Vidav–Palmer and
Gelfand–Naimark Theorems
155 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables (Second Edition)
156 L. W. Beineke and R. J. Wilson (eds.) with B. Toft Topics in Chromatic Graph Theory
157 T. Mora Solving Polynomial Equation Systems III: Algebraic Solving
158 T. Mora Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond
159 V. Berthé and M. Rigo (eds.) Combinatorics, Words and Symbolic Dynamics
160 B. Rubin Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis
161 M. Ghergu and S. D. Taliaferro Isolated Singularities in Partial Differential Inequalities
162 G. Molica Bisci, V. D. Radulescu and R. Servadei Variational Methods for Nonlocal Fractional Problems
163 S. Wagon The Banach–Tarski Paradox (Second Edition)
164 K. Broughan Equivalents of the Riemann Hypothesis I: Arithmetic Equivalents
165 K. Broughan Equivalents of the Riemann Hypothesis II: Analytic Equivalents
166 M. Baake and U. Grimm (eds.) Aperiodic Order II: Crystallography and Almost Periodicity
167 M. Cabrera García and Á. Rodríguez Palacios Non-Associative Normed Algebras II: Representation
Theory and the Zel’manov Approach
168 A. Yu. Khrennikov, S. V. Kozyrev and W. A. Zúñiga-Galindo Ultrametric Pseudodifferential Equations
and Applications
Encyclopedia of Mathematics and its Applications
Ultrametric Pseudodifferential
Equations and Applications
ANDREI YU. KHRENNIKOV
Linnéuniversitetet, Sweden
SERGEI V. KOZYREV
Steklov Institute of Mathematics, Moscow
W. A. ZÚÑIGA-GALINDO
Centro de Investigación y de Estudios Avanzados
del Instituto Politécnico Nacional, Mexico
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It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781107188822
DOI: 10.1017/9781316986707
© Cambridge University Press 2018
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2018
Printed in the United Kingdom by Clays, St Ives plc
A catalogue record for this publication is available from the British Library
ISBN 978-1-107-18882-2 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
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accurate or appropriate.
Dedicated to Vasilii Sergeevich Vladimirov
Contents
Preface page xi
1 p-Adic Analysis: Essential Ideas and Results 1
1.1 The Field of p-Adic Numbers 1
N
1.2 Topology of Qp 2
1.3 The Bruhat–Schwartz Space and the Fourier Transform 3
1.4 Distributions 4
1.5 Some Function Spaces 6
2 Ultrametric Geometry: Cluster Networks and Buildings 8
2.1 Introduction 8
2.2 Clustering, Trees, and Ultrametric Spaces 9
2.3 Family of Metrics and Multiclustering 12
2.4 Afine Bruhat–Tits Buildings and Cluster Networks 13
2.5 Groups Acting on Trees and the Vladimirov Operator 17
3 p-Adic Wavelets 20
3.1 Introduction 20
3.2 Basis of p-Adic Wavelets 26
3.3 Coherent States 28
3.4 Orbits of Mean-Zero Test Functions as Wavelet Frames 30
3.5 Multidimensional Wavelets and Representation Theory 33
3.6 Wavelets with Matrix Dilations 34
3.7 Wavelet Transform of Distributions 40
3.8 Relation to the Haar Basis on the Real Line 41
3.9 p-Adic Multiresolution Analysis 43
3.10 p-Adic One-Dimensional Haar Wavelet Bases 45
3.11 p-Adic Scaling Functions 48
3.12 Multiresolution Frames of Wavelets 49
3.13 Multidimensional Multiresolution Wavelet Bases 51
vii
viii Contents
3.14 p-Adic Shannon–Kotelnikov Theorem 53
3.15 Spectral Theory of p-Adic Pseudodifferential Operators 54
3.16 Wavelets and Operators for General Ultrametric Spaces 59
4 Ultrametricity in the Theory of Complex Systems 63
4.1 Introduction 63
4.2 p-Adic Parametrization of the Parisi Matrix 65
4.3 Dynamics on Complex Energy Landscapes 67
4.4 Actomyosin Molecular Motor 70
4.5 2-Adic Model of the Genetic Code 73
5 Some Applications of Wavelets and Integral Operators 76
5.1 Pseudodifferential Equations 76
5.2 Non-linear Equations and the Cascade Model of Turbulence 78
5.3 p-Adic Brownian Motion 81
6 p-Adic and Ultrametric Models in Geophysics 83
6.1 Tree-like Structures in Nature 84
6.2 p-Adic Coniguration Space for Networks of Capillaries and
Balance Equations for Densities of Fluids 85
6.3 Non-linear p-Adic Dynamics 89
7 Recent Development of the Theory of p-Adic Dynamical
Systems 94
7.1 Van der Put Series and Coordinate Representations of
Dynamical Maps 96
7.2 Recent Results about Measure-Preserving Functions and
Ergodic Dynamics 99
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions 105
8 Parabolic-Type Equations, Markov Processes, and Models of
Complex Hierarchical Systems 114
8.1 Introduction 114
8.2 Operators W, Parabolic-Type Equations, and Markov
Processes 115
8.3 Elliptic Pseudodifferential Operators, Parabolic-Type
Equations and Markov Processes 121
8.4 Non-Archimedean Reaction–Ultradiffusion Equations and
Complex Hierarchic Systems 123
9 Stochastic Heat Equation Driven by Gaussian Noise 133
9.1 Introduction 133
9.2 p-Adic Parabolic-Type Pseudodifferential Equations 134
9.3 Positive-Deinite Distributions and the Bochner–Schwartz
Theorem 136
9.4 Stochastic Integrals and Gaussian Noise 138
Contents ix
9.5 Stochastic Pseudodifferential Equations Driven by a
Spatially Homogeneous Noise 148
10 Sobolev-Type Spaces and Pseudodifferential Operators 155
10.1 Introduction 155
10.2 The Spaces H∞ 156
10.3 A Hörmander–Łojasiewicz-Type Estimation 162
10.4 The Spaces W∞ 165
10.5 Pseudodifferential Operators on W∞ 168
10.6 Existence of Fundamental Solutions 170
10.7 Igusa’s Local Zeta Functions and Fundamental Solutions 172
10.8 Local Zeta Functions and Pseudodifferential Operators in H∞ 175
11 Non-Archimedean White Noise, Pseudodifferential Stochastic
Equations, and Massive Euclidean Fields 177
11.1 Introduction 177
11.2 Preliminaries 178
11.3 Pseudodifferential Operators and Green Functions 179
11.4 The Generalized White Noise 187
11.5 Euclidean Random Fields as Convoluted Generalized White
Noise 190
N
11.6 The p-Adic Brownian Sheet on Q 195
p
12 Heat Traces and Spectral Zeta Functions for p-Adic Laplacians 198
12.1 Introduction 198
12.2 A Class of p-Adic Laplacians 199
12.3 Lizorkin Spaces, Eigenvalues, and Eigenfunctions for Aβ
Operators 201
12.4 Heat Traces and p-Adic Heat Equations on the Unit Ball 203
12.5 Analytic Continuation of Spectral Zeta Functions 210
References 214
Index 236
