Stochastic cauchy problems in infinite dimensions : generalized and regularized solutions

299 Pages·2016·3.446 MB·English

by  Melʹnikova, Irina Valerianovna

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
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Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized
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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