Stochastic cauchy problems in infinite dimensions : generalized and regularized solutions PDF
Preview Stochastic cauchy problems in infinite dimensions : generalized and regularized solutions
Stochastic Cauchy Problems in Infinite Dimensions Generalized and Regularized Solutions © 2016 by Taylor & Francis Group, LLC MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS Stochastic Cauchy Problems in Infinite Dimensions Generalized and Regularized Solutions Irina V. Melnikova © 2016 by Taylor & Francis Group, LLC MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS Series Editors John A. Burns Thomas J. Tucker Miklos Bona Michael Ruzhansky Published Titles Application of Fuzzy Logic to Social Choice Theory, John N. Mordeson, Davender S. Malik and Terry D. Clark Blow-up Patterns for Higher-Order: Nonlinear Parabolic, Hyperbolic Dispersion and Schrödinger Equations, Victor A. Galaktionov, Enzo L. Mitidieri, and Stanislav Pohozaev Complex Analysis: Conformal Inequalities and the Bieberbach Conjecture, Prem K. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2016 by Taylor & Francis Group, LLC To my family Boris, Alexandra, and Nikolai © 2016 by Taylor & Francis Group, LLC Contents Preface ix Introduction xi Symbol Description xvii I Well-Posed and Ill-Posed Abstract Cauchy Problems: The Concept of Regularization 1 1 Semi-groupmethodsforconstructionofexact,approximated, and regularized solutions 3 1.1 The Cauchy problem and strongly continuous semi-groups of solution operators . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The Cauchy problem with generators of regularized semi- groups: integrated, convoluted, and R-semi-groups . . . . . . 14 1.3 R-semi-groups and regularizing operators in the construction of approximated solutions to ill-posed problems . . . . . . . 36 2 Distribution methods for construction of generalized solutions to ill-posed Cauchy problems 43 2.1 Solutions in spaces of abstract distributions . . . . . . . . . . 44 2.2 Solutions in spaces of abstract ultra-distributions . . . . . . 54 2.3 Solutions to the Cauchy problem for differential systems in Gelfand–Shilov spaces . . . . . . . . . . . . . . . . . . . . . . 59 3 Examples. Supplements 75 3.1 Examples of regularizedsemi-groups and their generators . . 75 3.2 Examples of solutions to Petrovsky correct, conditionally correct, and incorrect systems . . . . . . . . . . . . . . . . . 84 3.3 Definitions and properties of spaces of test functions . . . . . 93 3.4 Generalized Fourier and Laplace transforms. Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 vii © 2016 by Taylor & Francis Group, LLC viii Contents II Infinite-Dimensional Stochastic Cauchy Problems 111 4 Weak, regularized, and mild solutions to Itˆo integrated stochastic Cauchy problems in Hilbert spaces 113 4.1 Hilbert space-valued variables, processes, and stochastic inte- grals. Main properties and results . . . . . . . . . . . . . . . 114 4.2 SolutionstoCauchyproblemsforequationswithadditivenoise and generators of regularizedsemi-groups . . . . . . . . . . . 138 4.3 Solutions to Cauchy problems for semi-linear equations with multiplicative noise . . . . . . . . . . . . . . . . . . . . . . . 166 4.4 ExtensionoftheFeynman–Kactheoremtothecaseofrelations between stochastic equations and PDEs in Hilbert spaces . . 179 5 Infinite-dimensional stochastic Cauchy problems with white noise processes in spaces of distributions 197 5.1 GeneralizedsolutionstolinearstochasticCauchyproblemswith generators of regularized semi-groups . . . . . . . . . . . . . 198 5.2 Quasi-linearstochasticCauchyprobleminabstractColombeau spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6 Infinite-dimensional extension of white noise calculus with application to stochastic problems 229 6.1 Spaces of Hilbert space-valued generalized random variables: ( ) (H). Basic examples . . . . . . . . . . . . . . . . . . . . 230 ρ 6.2 ASna−lysis of ( ) (H)-valued processes . . . . . . . . . . . . . 241 ρ S − 6.3 S-transform and Wick product. Hitsuda–Skorohod integral. Main properties. Connection with Itˆo integral . . . . . . . . 247 6.4 Generalized solutions to stochastic Cauchy problems in spaces of abstract stochastic distributions . . . . . . . . . . . . . . . 253 Bibliography 275 Index 283 © 2016 by Taylor & Francis Group, LLC Preface In recent decades there has been growing realization that elements of chance play an essential role in many processes around us, including processes in physics, biology, and finance. Mathematical models that give an accurate de- scriptionofthese processesleadto stochasticequationsin finite- andinfinite- dimensional spaces. So far most of the literature on stochastic equations has been focused on the finite-dimensional case. This book is devoted to stochastic differential equations for random pro- cesseswithvaluesinHilbertspaces.ThemainobjectisthestochasticCauchy problem X (t)=AX(t)+F(t,X)+B(t,X)W(t), t [0,T], X(0)=ζ, (P.1) ′ ∈ where A is the generator of a semi-group of operators in a Hilbert space H, W is a white noise with values in another Hilbert space H, B is an operator from H to H, and F is a non-linear term. Due to the well-known irregularity of white noise, the Cauchy problem (P.1) is usually replaced with the related integral equation by constructing the stochastic integral with respect to a Wiener process W, the “primitive” of W. Problems of this type with a “good” operator A that generates a C - 0 semi-group have been extensively studied in the literature. In this book, we consider a much wider class of operators A, namely, the operators that do not necessarily generate C -semi-groups, but generate 0 regularized semi-groups. Typical examples include generators of integrated, convoluted,andR-semi-groups.Moreover,alongwiththe“classical”approach to stochastic problems, which consists of solving the corresponding integral equations,weconsidertheCauchyprobleminitsinitialform(P.1)withwhite noise processes in spaces of distributions and obtain generalized solutions. Themotivationforwritingthebookwastwo-fold.First,togiveanaccount of modern semi-group and distribution methods in their interrelations with the methods of infinite-dimensional stochastic analysis, accessible to nonspe- cialists. Second, to show how the idea of regularization, which we treat as the regularizationin a broadsense, runs through all these methods. We hope that this idea will be useful for numerical realization and applications of the theory. The statedobjectives are implemented in two parts of the book. In PartI we give a self-contained introduction to modern semi-groupand abstract dis- tribution methods for solving the homogeneous (deterministic) Cauchy prob- lem.Wediscussbasicpropertiesofregularizedsemi-groupsandillustratethem ix © 2016 by Taylor & Francis Group, LLC x Preface with numerous examples, paying special attention to differential systems in Gelfand–Shilov spaces. In Part II the semi-group and distribution methods are used for solving stochastic problems along with the methods of infinite- dimensional stochastic analysis. This part also includes novel material that extendsthewhitenoiseanalysistoHilbertspacesandallowsustoobtainnew types of solutions to stochastic problems. IbeganmycareerinmathematicsasagraduatestudentofProf.ValentinK. Ivanov, one of the founders of the theory of ill-posed problems. I am grateful to him for his help and encouragement over the years. IwouldliketothankDr.AlexeiFilinkovforlong-termcooperation.During my visits to the University of Adelaide we wrote our first joint book for CRC Press,The AbstractCauchy Problem: Three Approaches,andplannedtowrite another one on stochastic problems. Unfortunately, Alexei had to withdraw from this project. It is my pleasure to thank my colleagues and friends Profs. Edward J. Allen, Jean Francois Colombeau, Angelo Favini, Andrzej Kaminski, Michael Oberguggenberger, Stevan Pilipovi´c, and Dora Seleˇsi for many useful discus- sions of problems related to the topic of the book. Last but not least, I am grateful to the members of my group at Ural FederalUniversity:Drs. Uliana Alekseeva,MaximAlshanskiy,Valentina Par- fenenkova,andpost-graduatesVadimBovkunandOlgaStarkova.Iowemuch to them in preparing this book. However, the final responsibility for the con- tent of this book lies solely with me. Ekaterinburg Ural Federal University Irina V. Melnikova © 2016 by Taylor & Francis Group, LLC
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