Random Perturbation Methods with Applications in Science and Engineering

501 Pages·2002·2.663 MB·English

by  Anatoli V. Skorokhod, Frank C. Hoppensteadt, Habib Salehi (eds.)

Random Perturbation
Methods with
Applications in
Science and Engineering
Anatoli V. Skorokhod
Frank C. Hoppensteadt
Habib Salehi
Springer
Applied Mathematical Sciences
Volume 150
Editors
S.S. Antman J.E. Marsden L. Sirovich
Advisors
J.K. Hale P. Holmes J. Keener
J. Keller B.J. Matkowsky A. Mielke
C.S. Peskin K.R.S. Sreenivasan
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Anatoli V. Skorokhod Frank C. Hoppensteadt
Habib Salehi
Random Perturbation Methods
with Applications in
Science and Engineering
With 31 Figures
AnatoliV.Skorokhod FrankC.Hoppensteadt HabibSalehi
InstituteofMathematics SystemsScienceand DepartmentofStatistics
UkrainianAcademyof EngineeringResearch andProbability
Science Center MichiganStateUniversity
3TereshchenkivskaStreet ArizonaStateUniversity EastLansing,MI48824
Kiev01601,Ukraine Tempe,AZ85287-7606 USA
and USA [email protected]
DepartmentofStatistics [email protected]
andProbability
MichiganStateUniversity
EastLansing,MI48824
USA
[email protected]
Editors
S.S.Antman J.E.Marsden L.Sirovich
DepartmentofMathematics ControlandDynamical DivisionofApplied
and Systems,107-81 Mathematics
InstituteforPhysicalScience CaliforniaInstituteof BrownUniversity
andTechnology Technology Providence,RI02912
UniversityofMaryland Pasadena,CA91125 USA
CollegePark,MD20742-4015 USA [email protected]
USA [email protected]
[email protected]
MathematicsSubjectClassification(2000):37-02,60JXX,70K50
LibraryofCongressCataloging-in-PublicationData
Skorokhod,A.V.(AnatoliiVladimirovich),1930–
Randomperturbationmethodswithapplicationsinscienceandengineering/Anatoli
V.Skorokhod,FrankC.Hoppensteadt,HabibSalehi.
p.cm.—(Appliedmathematicalsciences;150)
Includesbibliographicalreferencesandindex.
ISBN0-387-95427-9(acid-freepaper)
1.Perturbation(Mathematics) 2.Differentiabledynamicalsystems. I.Hoppensteadt,
F.C. II.Salehi,Habib. III.Title. IV.Appliedmathematicalsciences(Springer-Verlag
NewYork,Inc.);v.150.
QA1.A647vol.150b
[QA871]
510s—dc21
[515′.35] 2001059799
ISBN0-387-95427-9 Printedonacid-freepaper.
2002Springer-VerlagNewYork,Inc.
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Preface
This book has its roots in two different areas of mathematics: pure
mathematics,wherestructuresarediscoveredinthecontextofothermath-
ematical structures and investigated, and applications of mathematics,
where mathematical structures are suggested by real–world problems aris-
ing in science and engineering, investigated, and then used to address the
motivating problem. While there are philosophical differences between ap-
pliedandpuremathematicalscientists,itisoftendifficulttosortthemout.
The authors of this book reflect these different approaches.
ThisprojectbeganwhenProfessorSkorokhodoftheUkranianAcademy
of Sciences joined the faculty in the Department of Statistics and Prob-
ability at Michigan State University in 1993. After that, the authors
collaborated on numerous joint publications that have culminated in the
production of this book.
The structure of the book is roughly in two parts: The first part (Chap-
ters 1–7) presents a careful development of mathematical methods needed
to study random perturbations of dynamical systems. The second part
(Chapters 8–12) presents nonrandom problems in a variety of important
applications, reformulations of them that account for both external and
system random noise, and applications of the results from the first part to
analyze,simulate,andvisualizetheseproblemsperturbedbynoise.Inmost
cases,weidentifywhichresultsarenovelandwhicharecloselyrelatedtode-
velopmentsbyothers.Wehavetriedtoacknowledgetheprimarysourceson
which our work is based, but we apologize for those we have inadvertently
omitted.
The authors are grateful for the support in these projects from the De-
partmentofStatisticsandProbabilityandtheCollegeofNaturalScienceat
Michigan State University, the Systems Science and Engineering Research
Center at Arizona State University, and the National Science Foundation.
Kiev, Ukraine Anatoli V. Skorokhod
Paradise Valley, Arizona, USA Frank C. Hoppensteadt
East Lansing, Michigan, USA Habib Salehi
December 2001
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Contents
Preface v
Introduction 1
What Are Dynamical Systems? . . . . . . . . . . . . . . . . . . 1
What Is Random Noise? . . . . . . . . . . . . . . . . . . . . . . 3
What Are Ergodic Theorems? . . . . . . . . . . . . . . . . . . . 8
What Happens for t Large? . . . . . . . . . . . . . . . . . . . . 18
What Is in This Book? . . . . . . . . . . . . . . . . . . . . . . . 21
1 Ergodic Theorems 49
1.1 Birkhoff’s Classical Ergodic Theorem . . . . . . . . . . . 49
1.1.1 Mixing Conditions . . . . . . . . . . . . . . . . . 53
1.1.2 Discrete–Time Stationary Processes . . . . . . . . 54
1.2 Discrete–Time Markov Processes . . . . . . . . . . . . . 56
1.3 Continuous–Time Stationary Processes . . . . . . . . . . 60
1.4 Continuous–Time Markov Processes . . . . . . . . . . . . 61
2 Convergence Properties of Stochastic Processes 64
2.1 Weak Convergence of Stochastic Processes . . . . . . . . 64
2.1.1 Weak Compactness in C . . . . . . . . . . . . . . 66
2.2 Convergence to a Diffusion Process . . . . . . . . . . . . 68
2.2.1 Diffusion Processes . . . . . . . . . . . . . . . . . 68
2.2.2 Weak Convergence to a Diffusion Process. . . . . 70
2.3 Central Limit Theorems for Stochastic Processes . . . . 73
viii Contents
2.3.1 Continuous–Time Markov Processes. . . . . . . . 73
2.3.2 Discrete–Time Markov Processes . . . . . . . . . 75
2.3.3 Discrete–Time Stationary Processes . . . . . . . . 76
2.3.4 Continuous–Time Stationary Processes . . . . . . 79
2.4 Large Deviation Theorems . . . . . . . . . . . . . . . . . 80
2.4.1 Continuous–Time Markov Processes. . . . . . . . 81
2.4.2 Discrete–Time Markov Processes . . . . . . . . . 86
3 Averaging 88
3.1 Volterra Integral Equations. . . . . . . . . . . . . . . . . 88
3.1.1 Linear Volterra Integral Equations . . . . . . . . 92
3.1.2 Some Nonlinear Equations . . . . . . . . . . . . . 97
3.2 Differential Equations . . . . . . . . . . . . . . . . . . . . 99
3.2.1 Linear Differential Equations. . . . . . . . . . . . 101
3.3 Difference Equations . . . . . . . . . . . . . . . . . . . . 102
3.3.1 Linear Difference Equations . . . . . . . . . . . . 105
3.4 Large Deviation for Differential Equations . . . . . . . . 105
3.4.1 Some Auxiliary Results . . . . . . . . . . . . . . . 106
3.4.2 Main Theorem. . . . . . . . . . . . . . . . . . . . 110
3.4.3 Systems with Additive Perturbations . . . . . . . 112
4 Normal Deviations 114
4.1 Volterra Integral Equations. . . . . . . . . . . . . . . . . 114
4.2 Differential Equations . . . . . . . . . . . . . . . . . . . . 120
4.2.1 Markov Perturbations . . . . . . . . . . . . . . . 127
4.3 Difference Equations . . . . . . . . . . . . . . . . . . . . 128
5 Diffusion Approximation 133
5.1 Differential Equations . . . . . . . . . . . . . . . . . . . . 133
5.1.1 Markov Jump Perturbations . . . . . . . . . . . . 134
5.1.2 Some Generalizations . . . . . . . . . . . . . . . . 140
5.1.3 General Markov Perturbations . . . . . . . . . . . 145
5.1.4 Stationary Perturbations . . . . . . . . . . . . . . 146
5.1.5 Diffusion Approximations to First Integrals . . . 150
5.2 Difference Equations . . . . . . . . . . . . . . . . . . . . 156
5.2.1 Markov Perturbations . . . . . . . . . . . . . . . 156
5.2.2 Diffusion Approximations to First Integrals . . . 161
5.2.3 Stationary Perturbations . . . . . . . . . . . . . . 166
6 Stability 172
6.1 Stability of Perturbed Differential Equations . . . . . . . 172
6.1.1 Jump Perturbations of Nonlinear Equations . . . 173
6.1.2 Stationary Perturbations . . . . . . . . . . . . . . 182
6.2 Stochastic Resonance for Gradient Systems . . . . . . . . 193
6.2.1 Large Deviations near a Stable Static State . . . 193
Contents ix
6.2.2 Transitions Between Stable Static States . . . . . 198
6.2.3 Stochastic Resonance . . . . . . . . . . . . . . . 199
6.3 Randomly Perturbed Difference Equations . . . . . . . . 200
6.3.1 Markov Perturbations: Linear Equations . . . . . 201
6.3.2 Stationary Perturbations . . . . . . . . . . . . . . 203
6.3.3 Markov Perturbations: Nonlinear Equations . . . 205
6.3.4 Stationary Perturbations . . . . . . . . . . . . . . 210
6.3.5 Small Perturbations of a Stable System . . . . . . 211
6.4 Convolution Integral Equations . . . . . . . . . . . . . . 216
6.4.1 Laplace Transforms and Their Inverses . . . . . . 218
6.4.2 Laplace Transforms of Noisy Kernels . . . . . . . 222
7 Markov Chains with Random Transition Probabilities 232
7.1 Stationary Random Environment . . . . . . . . . . . . . 233
7.2 Weakly Random Environments . . . . . . . . . . . . . . 243
7.3 Markov Processes with Randomly Perturbed
Transition Probabilities . . . . . . . . . . . . . . . . . . . 249
7.3.1 Stationary Random Environments . . . . . . . . . 249
7.3.2 Ergodic Theorem for Markov Processes
in Random Environments . . . . . . . . . . . . . 253
7.3.3 Markov Process in a Weakly Random
Environment . . . . . . . . . . . . . . . . . . . . . 254
8 Randomly Perturbed Mechanical Systems 257
8.1 Conservative Systems with Two Degrees of Freedom. . . 257
8.1.1 Conservative Systems . . . . . . . . . . . . . . . . 258
8.1.2 Randomly Perturbed Conservative Systems . . . 262
8.1.3 Behavior of the Perturbed System
near a Knot . . . . . . . . . . . . . . . . . . . . . 273
8.1.4 Diffusion Processes on Graphs . . . . . . . . . . . 285
8.1.5 Simulation of a Two-Well Potential Problem . . . 290
8.2 Linear Oscillating Conservative Systems . . . . . . . . . 290
8.2.1 Free Linear Oscillating Conservative Systems . . 290
8.2.2 Randomly Perturbed Linear
Oscillating Systems . . . . . . . . . . . . . . . . . 293
8.3 A Rigid Body with a Fixed Point . . . . . . . . . . . . . 297
8.3.1 Motion of a Rigid Body around
a Fixed Point . . . . . . . . . . . . . . . . . . . . 298
8.3.2 Analysis of Randomly Perturbed Motions . . . . 299
9 Dynamical Systems on a Torus 303
9.1 Theory of Rotation Numbers . . . . . . . . . . . . . . . . 303
9.1.1 Existence of the Rotation Number . . . . . . . . 305
9.1.2 Purely Periodic Systems . . . . . . . . . . . . . . 307
9.1.3 Ergodic Systems. . . . . . . . . . . . . . . . . . . 308