Taylor Approximations for Stochastic Partial Differential Equations (CBMS-NSF Regional Conference Series in Applied Mathematics)

235 Pages·2010·1.53 MB·English

by  Arnulf Jentzen and Peter Kloeden

Taylor Approximations
for Stochastic Partial
Differential Equations
cB83_Jentzen_fM.indd 1 9/28/2011 3:28:57 PM
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CBMS-NSF REGIONAL CONFERENCE SERIES
IN APPLIED MATHEMATICS
A series of lectures on topics of current research interest in applied mathematics under the direction of
the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and
published by SIAM.
Garrett Birkhoff, The Numerical Solution of Elliptic Equations
D. V. Lindley, Bayesian Statistics, A Review
R. S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis
R. R. Bahadur, Some Limit Theorems in Statistics
Patrick Billingsley, Weak Convergence of Measures: Applications in Probability
J. L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems
Roger Penrose, Techniques of Differential Topology in Relativity
Herman Chernoff, Sequential Analysis and Optimal Design
J. Durbin, Distribution Theory for Tests Based on the Sample Distribution Function
Sol I. Rubinow, Mathematical Problems in the Biological Sciences
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves
I. J. Schoenberg, Cardinal Spline Interpolation
Ivan Singer, The Theory of Best Approximation and Functional Analysis
Werner C. Rheinboldt, Methods of Solving Systems of Nonlinear Equations
Hans F. Weinberger, Variational Methods for Eigenvalue Approximation
R. Tyrrell Rockafellar, Conjugate Duality and Optimization
Sir James Lighthill, Mathematical Biofluiddynamics
Gerard Salton, Theory of Indexing
Cathleen S. Morawetz, Notes on Time Decay and Scattering for Some Hyperbolic Problems
F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics
Richard Askey, Orthogonal Polynomials and Special Functions
L. E. Payne, Improperly Posed Problems in Partial Differential Equations
S. Rosen, Lectures on the Measurement and Evaluation of the Performance of Computing Systems
Herbert B. Keller, Numerical Solution of Two Point Boundary Value Problems
J. P. LaSalle, The Stability of Dynamical Systems
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications
Peter J. Huber, Robust Statistical Procedures
Herbert Solomon, Geometric Probability
Fred S. Roberts, Graph Theory and Its Applications to Problems of Society
Juris Hartmanis, Feasible Computations and Provable Complexity Properties
Zohar Manna, Lectures on the Logic of Computer Programming
Ellis L. Johnson, Integer Programming: Facets, Subadditivity, and Duality for Group and
Semi-Group Problems
Shmuel Winograd, Arithmetic Complexity of Computations
J. F. C. Kingman, Mathematics of Genetic Diversity
Morton E. Gurtin, Topics in Finite Elasticity
Thomas G. Kurtz, Approximation of Population Processes
Jerrold E. Marsden, Lectures on Geometric Methods in Mathematical Physics
Bradley Efron, The Jackknife, the Bootstrap, and Other Resampling Plans
M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis
D. H. Sattinger, Branching in the Presence of Symmetry
R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis
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Miklós Csörgo, Quantile Processes with Statistical Applications
J. D. Buckmaster and G. S. S. Ludford, Lectures on Mathematical Combustion
R. E. Tarjan, Data Structures and Network Algorithms
Paul Waltman, Competition Models in Population Biology
S. R. S. Varadhan, Large Deviations and Applications
Kiyosi Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces
Alan C. Newell, Solitons in Mathematics and Physics
Pranab Kumar Sen, Theory and Applications of Sequential Nonparametrics
László Lovász, An Algorithmic Theory of Numbers, Graphs and Convexity
E. W. Cheney, Multivariate Approximation Theory: Selected Topics
Joel Spencer, Ten Lectures on the Probabilistic Method
Paul C. Fife, Dynamics of Internal Layers and Diffusive Interfaces
Charles K. Chui, Multivariate Splines
Herbert S. Wilf, Combinatorial Algorithms: An Update
Henry C. Tuckwell, Stochastic Processes in the Neurosciences
Frank H. Clarke, Methods of Dynamic and Nonsmooth Optimization
Robert B. Gardner, The Method of Equivalence and Its Applications
Grace Wahba, Spline Models for Observational Data
Richard S. Varga, Scientific Computation on Mathematical Problems and Conjectures
Ingrid Daubechies, Ten Lectures on Wavelets
Stephen F. McCormick, Multilevel Projection Methods for Partial Differential Equations
Harald Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods
Joel Spencer, Ten Lectures on the Probabilistic Method, Second Edition
Charles A. Micchelli, Mathematical Aspects of Geometric Modeling
Roger Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, Second Edition
Glenn Shafer, Probabilistic Expert Systems
Peter J. Huber, Robust Statistical Procedures, Second Edition
J. Michael Steele, Probability Theory and Combinatorial Optimization
Werner C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, Second Edition
J. M. Cushing, An Introduction to Structured Population Dynamics
Tai-Ping Liu, Hyperbolic and Viscous Conservation Laws
Michael Renardy, Mathematical Analysis of Viscoelastic Flows
Gérard Cornuéjols, Combinatorial Optimization: Packing and Covering
Irena Lasiecka, Mathematical Control Theory of Coupled PDEs
J. K. Shaw, Mathematical Principles of Optical Fiber Communications
Zhangxin Chen, Reservoir Simulation: Mathematical Techniques in Oil Recovery
Athanassios S. Fokas, A Unified Approach to Boundary Value Problems
Margaret Cheney and Brett Borden, Fundamentals of Radar Imaging
Fioralba Cakoni, David Colton, and Peter Monk, The Linear Sampling Method in
Inverse Electromagnetic Scattering
Adrian Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions
and Tsunamis
Wei-Ming Ni, The Mathematics of Diffusion
Arnulf Jentzen and Peter E. Kloeden, Taylor Approximations for Stochastic Partial Differential
Equations
cB83_Jentzen_fM.indd 3 9/28/2011 3:28:57 PM
Arnulf J EnTzEn
Princeton university
Princeton, new Jersey
PETEr E. Klo EDEn
Goethe university
f rankfurt am Main, Germany
Taylor Approximations
for Stochastic Partial
Differential Equations
Soci ETy for inDuSTri Al AnD APPli ED MAThEMATicS
Phil ADEl PhiA
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Copyright © 2011 by the Society for Industrial and Applied Mathematics.
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of this book may
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Figure 3.1 and 6.1 used with permission from Springer.
Figure 4.1 used with permission from Cambridge University Press.
Figures 8.1-8.10 used with permission from the American Institute of Mathematical
Sciences.
Library of Congress Cataloging-in-Publication Data
Jentzen, Arnulf.
Taylor approximations for stochastic partial differential equations / Arnulf Jentzen, Peter E.
Kloeden.
p. cm. -- (CBMS-NSF regional conference series in applied mathematics ; 83)
Includes bibliographical references and index.
ISBN 978-1-611972-00-9
1. Stochastic partial differential equations. 2. Approximation theory. I. Kloeden, Peter E.
II. Title.
QA274.25.J46 2011
515'.353--dc23
2011029546

is a registered trademark.
cB83_Jentzen_fM.indd 6 9/28/2011 3:28:58 PM
Contents
Preface xi
ListofFigures xiii
1 Introduction 1
1.1 TaylorexpansionsforODEs . . . . . . . . . . . . . . . . . . . 1
1.1.1 TaylorschemesforODEs . . . . . . . . . . . . . 3
1.1.2 IntegralrepresentationofTaylorexpansions . . . 4
I RandomandStochasticOrdinaryDifferentialEquations 7
2 TaylorExpansionsandNumericalSchemesforRODEs 9
2.1 RODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 EquivalenceofRODEsandSODEs . . . . . . . . 10
2.1.2 SimplenumericalschemesforRODEs . . . . . . 12
2.2 Taylor-likeexpansionsforRODEs . . . . . . . . . . . . . . . 13
2.2.1 Multi-indexnotation . . . . . . . . . . . . . . . . 14
2.2.2 Taylorexpansionsofthevectorfield . . . . . . . 14
2.3 RODE–Taylorschemes . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Discretizationerror . . . . . . . . . . . . . . . . 17
2.3.2 ExamplesofRODE–Taylorschemes . . . . . . . 22
3 SODEs 25
3.1 ItôSODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Existence and uniqueness of strong solutions of
SODEs . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 SimplenumericalschemesforSODEs . . . . . . 27
3.2 Itô–Taylorexpansions . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 IteratedapplicationoftheItôformula . . . . . . . 29
3.2.2 GeneralstochasticTaylorexpansions . . . . . . . 31
3.3 Itô–TaylornumericalschemesforSODEs . . . . . . . . . . . 32
vii
viii Contents
3.4 Pathwiseconvergence . . . . . . . . . . . . . . . . . . . . . . 35
3.4.1 NumericalschemesforRODEsappliedtoSODEs 36
3.5 Restrictivenessofthestandardassumptions. . . . . . . . . . . 37
3.5.1 CounterexamplesfortheEuler–Maruyamascheme 37
4 NumericalMethodsforSODEswithNonstandardAssumptions 43
4.1 SODEswithoutuniformlyboundedcoefficients . . . . . . . . 43
4.2 SODEsonrestrictedregions. . . . . . . . . . . . . . . . . . . 44
4.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Anothertypeofweakconvergence . . . . . . . . . . . . . . . 47
II StochasticPartialDifferentialEquations 51
5 StochasticPartialDifferentialEquations 53
5.1 RandomandstochasticPDEs . . . . . . . . . . . . . . . . . . 53
5.1.1 MildsolutionsofSPDEs. . . . . . . . . . . . . . 54
5.2 Functionalanalyticalpreliminaries . . . . . . . . . . . . . . . 55
5.2.1 Hilbert–Schmidtandtrace-classoperators . . . . 55
5.2.2 Hilbertspacevaluedrandomvariables . . . . . . 56
5.2.3 Hilbertspacevaluedstochasticprocesses . . . . . 56
5.2.4 InfinitedimensionalWienerprocesses. . . . . . . 56
5.3 Settingandassumptions . . . . . . . . . . . . . . . . . . . . . 57
5.4 Existence,uniqueness,andregularityofsolutionsofSPDEs . . 58
5.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5.1 FinitedimensionalSODEs. . . . . . . . . . . . . 60
5.5.2 SecondorderSPDEs . . . . . . . . . . . . . . . . 60
5.5.3 FourthorderSPDEs . . . . . . . . . . . . . . . . 65
5.5.4 SPDEswithtime-dependentcoefficients . . . . . 67
6 NumericalMethodsforSPDEs 69
6.1 Anearlyresult . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Otherresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 TheexponentialEulerscheme . . . . . . . . . . . . . . . . . . 73
6.3.1 Convergence . . . . . . . . . . . . . . . . . . . . 74
6.3.2 Numericalresults . . . . . . . . . . . . . . . . . 75
6.3.3 Restrictivenessoftheassumptions . . . . . . . . 76
7 TaylorApproximationsforSPDEswithAdditiveNoise 79
7.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.1.1 Propertiesofthesolutions . . . . . . . . . . . . . 83
7.2 Autonomization . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.3.1 SemigroupgeneratedbytheLaplacian . . . . . . 91
Contents ix
7.3.2 ThedriftasaNemytskiioperator . . . . . . . . . 92
7.3.3 Stochasticprocessasstochasticconvolution . . . 93
7.3.4 ConcreteexamplesofSPDEswithadditivenoise . 94
7.4 Taylorexpansions . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4.1 Integraloperators . . . . . . . . . . . . . . . . . 98
7.5 AbstractexamplesofTaylorexpansions . . . . . . . . . . . . 104
7.6 ExamplesofTaylorapproximations . . . . . . . . . . . . . . . 114
7.6.1 Space–timewhitenoise . . . . . . . . . . . . . . 114
7.6.2 TaylorapproximationsforanonlinearSPDE . . . 116
7.6.3 Smoothernoise . . . . . . . . . . . . . . . . . . 121
7.7 NumericalschemesfromTaylorexpansions . . . . . . . . . . 121
7.7.1 TheexponentialEulerscheme . . . . . . . . . . . 123
7.7.2 ARunge–KuttaschemeforSPDEs . . . . . . . . 124
8 TaylorApproximationsforSPDEswithMultiplicativeNoise 127
8.1 HeuristicderivationofTaylorexpansions . . . . . . . . . . . . 127
8.2 Settingandassumptions . . . . . . . . . . . . . . . . . . . . . 130
8.2.1 Stochasticheatequation . . . . . . . . . . . . . . 133
8.3 TaylorexpansionsforSPDEs . . . . . . . . . . . . . . . . . . 135
8.3.1 Integraloperators . . . . . . . . . . . . . . . . . 136
8.3.2 DerivationofsimpleTaylorexpansions . . . . . . 138
8.3.3 HigherorderTaylorexpansions . . . . . . . . . . 139
8.4 Stochastictreesandwoods . . . . . . . . . . . . . . . . . . . 140
8.4.1 Stochastictreesandwoods . . . . . . . . . . . . 140
8.4.2 Constructionofstochastictreesandwoods . . . . 142
8.4.3 Subtrees . . . . . . . . . . . . . . . . . . . . . . 145
8.4.4 Orderofstochastictreesandwoods . . . . . . . . 146
8.4.5 StochasticwoodsandTaylorexpansions . . . . . 147
8.5 ExamplesofTaylorapproximations . . . . . . . . . . . . . . . 149
8.5.1 AbstractexamplesofTaylorapproximations . . . 149
8.5.2 Applicationtothestochasticheatequation . . . . 151
8.5.3 FinitedimensionalSODEs. . . . . . . . . . . . . 152
8.6 NumericalschemesforSPDEs . . . . . . . . . . . . . . . . . 154
8.6.1 TheexponentialEulerscheme . . . . . . . . . . . 155
8.6.2 AninfinitedimensionalanalogueofMilstein’s
scheme . . . . . . . . . . . . . . . . . . . . . . . 156
8.6.3 Linear-implicitEulerandCrank–Nicolson
schemes . . . . . . . . . . . . . . . . . . . . . . 157
8.6.4 Globalandlocalconvergenceorders . . . . . . . 157
8.6.5 Numericalsimulations . . . . . . . . . . . . . . . 158
8.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.7.1 ProofofLemma8.3 . . . . . . . . . . . . . . . . 160
8.7.2 ProofofLemma8.5 . . . . . . . . . . . . . . . . 163