Levy Processes, Integral Equations, Statistical Physics: Connections and Interactions

246 Pages·2012·3.596 MB·English

by  Lev A. Sakhnovich

Operator Theory: Advances and Applications
Volume 225

Founded in 1979 by Israel Gohberg

Editors:
Joseph A. Ball (Blacksburg, VA, USA)
Harry Dym (Rehovot, Israel)
Marinus A. Kaashoek (Amsterdam, The Netherlands)
Heinz Langer (Vienna, Austria)
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Associate Editors: Honorary and Advisory Editorial Board:
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Albrecht Böttcher (Chemnitz, Germany) Ciprian Foias (College Station, TX, USA)
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Subseries
Linear Operators and Linear Systems
Subseries editors:
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Birgit Jacob (Wuppertal, Germany)
André C.M. Ran (Amsterdam, The Netherlands)

Subseries
Advances in Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze (Potsdam, Germany)
Michael Demuth (Clausthal, Germany)
Jerome A. Goldstein (Memphis, TN, USA)
Nobuyuki Tose (Yokohama, Japan)
Ingo Witt (Göttingen, Germany)
Lev A. Sakhnovich
Levy Processes,
Integral Equations,
Statistical Physics:
Connections
and Interactions
Prof. Lev A. Sakhnovich
99 Cove Avenue
Milford, Connecticut
USA
ISBN 978-3-0348-0355-7 ISBN 978-3-0348-0356-4 (eBook)
DOI 10.1007/978-3-0348-0356-4
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To memory of my dear wife Olena Melnychenko

Contents
Introduction 1
1 Levy processes 11
1.1 Main notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Convolution type form of infinitesimal generator . . . . . . . . . . 13
1.3 Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Truncated generators and quasi-potentials . . . . . . . . . . . . . . 20
1.5 Probability of the Levy process remaining within the given domain 22
1.6 Non-negativity of the kernel Φ(x,y) . . . . . . . . . . . . . . . . . 25
1.7 Sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.8 Quasi-potential B, structure and properties . . . . . . . . . . . . . 30
1.9 Long time behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.10 Stable processes, main notions . . . . . . . . . . . . . . . . . . . . 39
1.11 Stable processes, quasi-potential . . . . . . . . . . . . . . . . . . . 40
1.12 On sample functions behavior of stable processes . . . . . . . . . . 42
1.13 Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.14 Iterated logarithm law, most visited sites and first hitting time . . 47
1.15 Two-sided estimation of the greatest eigenvalue of the operator B 51
α
2 The principle of imperceptibility of the boundary in the theory
of stable processes 53
2.1 On a probabilistic inequality . . . . . . . . . . . . . . . . . . . . . 53
2.2 A weakened principle of imperceptibility of the boundary . . . . . 55
2.3 Cauchy process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4 Wiener process, case α=2 . . . . . . . . . . . . . . . . . . . . . . 60
2.5 General case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Approximation of positive functions by linear positive polynomial
operators 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 The asymptotic formula for C (ϕ,α) . . . . . . . . . . . . . . . . . 64
n
3.3 Precise value of C∗(α) . . . . . . . . . . . . . . . . . . . . . . . . . 73
vii
viii Contents
3.4 Korovkin’s and Fejer’s operators . . . . . . . . . . . . . . . . . . . 74
4 Optimal prediction and matched filtering for generalized stationary
processes 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Generalized stationary processes . . . . . . . . . . . . . . . . . . . 79
4.3 Generalized processes, examples . . . . . . . . . . . . . . . . . . . . 80
4.4 Problem of optimal prediction . . . . . . . . . . . . . . . . . . . . . 82
4.5 Generalized matched filters . . . . . . . . . . . . . . . . . . . . . . 83
5 Effective construction of a class of positive operators in Hilbert space,
which do not admit triangular factorization 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 A special class of operators and corresponding differential systems 86
5.3 Non-factorable positive definite operators, a sufficient condition . . 90
5.4 A class of non-factorable positive definite operators . . . . . . . . . 92
5.5 Examples instead of existence theorems . . . . . . . . . . . . . . . 96
5.6 White noise type process, a special case . . . . . . . . . . . . . . . 97
6 Comparison of thermodynamic characteristics of quantum and
classical approaches 101
6.1 Quasi-classical case . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 One-dimensional potential well . . . . . . . . . . . . . . . . . . . . 103
6.3 Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 General case, statistical sum . . . . . . . . . . . . . . . . . . . . . 107
6.5 General case, mean energy . . . . . . . . . . . . . . . . . . . . . . . 111
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7 Dual canonical systems and dual matrix string equations 113
7.1 Canonical differential system . . . . . . . . . . . . . . . . . . . . . 116
7.2 On reduction of the canonical system to two dual differential
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3 Spectral data and uniqueness theorems . . . . . . . . . . . . . . . . 127
7.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.5 On a mean value theorem in the class of Nevanlinna functions and
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.6 Dual discrete canonical systems and dual orthogonal polynomials . 134
7.7 Classical discrete systems. Examples . . . . . . . . . . . . . . . . . 140
8 Integrable operators and canonical differential systems 151
8.1 Integrable operators and Riemann–Hilbert problem . . . . . . . . . 154
8.2 Limiting values of the multiplicative integral. . . . . . . . . . . . . 157
8.3 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.4 Canonical differential systems . . . . . . . . . . . . . . . . . . . . . 161
Contents ix
8.5 Inverse problem, examples . . . . . . . . . . . . . . . . . . . . . . . 163
9 The game between energy and entropy 169
9.1 Connection between energy and entropy (quantum case) . . . . . . 169
9.2 Connection between energy and entropy (classical case) . . . . . . 171
9.3 Third law of thermodynamics . . . . . . . . . . . . . . . . . . . . . 173
9.4 Entropy and energy in non-extensive statistical mechanics . . . . . 174
9.5 Algorithmic entropy, thermodynamics, and game interpretation . . 177
10 Inhomogeneous Boltzmann equations: distance, asymptotics and
comparison of the classical and quantum cases 181
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.2 Preliminaries: basic definitions and results . . . . . . . . . . . . . . 183
10.3 Extremal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
10.4 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.5 Modified Boltzmann equations for Fermi and Bose particles . . . . 189
10.6 Modified extremal problem . . . . . . . . . . . . . . . . . . . . . . 190
10.7 Lyapunov functional . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11 Operator Bezoutiant and roots of entire functions, concrete examples 201
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.2 Main notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
11.3 Properties of the operator B . . . . . . . . . . . . . . . . . . . . . 205
11.4 The explicit form of the Bezoutiant . . . . . . . . . . . . . . . . . . 208
11.5 Classes of entire functions without common zeroes . . . . . . . . . 212
11.6 A generalization of the Schur–Cohn theorem, examples . . . . . . . 218
Comments 223
Bibliography 227
Glossary of important notations 241
Index 243