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Stochastic Equations in Infinite Dimensions [Pehme köide]

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, (Polish Academy of Sciences)
  • Formaat: Paperback / softback, 476 pages, kõrgus x laius x paksus: 235x155x25 mm, kaal: 668 g
  • Sari: Encyclopedia of Mathematics and its Applications
  • Ilmumisaeg: 04-Feb-2008
  • Kirjastus: Cambridge University Press
  • ISBN-10: 0521059801
  • ISBN-13: 9780521059800
Teised raamatud teemal:
Stochastic Equations in Infinite DimensionsSuurem pilt
  • Formaat: Paperback / softback, 476 pages, kõrgus x laius x paksus: 235x155x25 mm, kaal: 668 g
  • Sari: Encyclopedia of Mathematics and its Applications
  • Ilmumisaeg: 04-Feb-2008
  • Kirjastus: Cambridge University Press
  • ISBN-10: 0521059801
  • ISBN-13: 9780521059800
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780521059800.html
The aim of this book is to give a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional spaces.

The aim of this book is to give a systematic and self-contained presentation of the basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. These are a generalization of stochastic differential equations as introduced by Itô and Gikhman that occur, for instance, when describing random phenomena that crop up in science and engineering, as well as in the study of differential equations. The book is divided into three parts. In the first the authors give a self-contained exposition of the basic properties of probability measures on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof.

Arvustused

"...very fine...provides the first comprehensive synthesis of the semigroup approach to SPDE....The exposition is excellent and readable throughout, and should help bring the theory to a wider audience." Daniel L. Ocone, Stochastics and Stochastics Reports "...this is an excellent book which covers a large part of stochastic evolution equations with clear proofs and a very interesting analysis of their properties...In my opinion this book will become an indispensable tool for everyone working on stochastic evolution equations and related areas." P. Kotelenez, Mathematical Reviews

Muu info

The aim of this book is to give a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional spaces.
Preface xiii
Introduction: Motivating examples 1(12)
0.1 Lifts of diffusion processes
1(1)
0.2 Markovian lifting of stochastic delay equations
2(1)
0.3 Zakai's equation
3(1)
0.4 Random motion of a string
4(2)
0.5 Stochastic equation of the free field
6(1)
0.6 Equation of stochastic quantization
6(3)
0.7 Reaction diffusion equation
9(1)
0.8 An example arising in neurophysiology
10(1)
0.9 Equation of population genetics
10(3)
I Foundations
13(102)
1 Random Variables
15(15)
1.1 Random variables and their integrals
15(8)
1.2 Operator valued random variables
23(4)
1.3 Conditional expectation and independence
27(3)
2 Probability measures
30(40)
2.1 General properties
30(6)
2.2 Gaussian measures in Banach spaces
36(12)
2.2.1 Fernique theorem
36(4)
2.2.2 Reproducing kernels
40(3)
2.2.3 White noise expansions
43(5)
2.3 Probability measures in Hilbert spaces
48(22)
2.3.1 Bochner theorem
48(5)
2.3.2 Gaussian measures on Hilbert spaces
53(5)
2.3.3 Feldman and Hajek theorem
58(10)
2.3.4 An application to the general Cameron-Martin formula
68(2)
3 Stochastic processes
70(16)
3.1 General concepts
70(2)
3.2 Kolmogorov test
72(3)
3.3 Processes with filtration
75(2)
3.4 Martingales
77(5)
3.5 Stopping times and Markov processes
82(1)
3.6 Gaussian processes in Hilbert spaces
83(1)
3.7 Stochastic processes as random variables
84(2)
4 The stochastic integral
86(29)
4.1 Hilbert space valued Wiener processes
86(4)
4.2 Definition of the stochastic integral
90(6)
4.3 Stochastic integral for cylindrical Wiener processes
96(5)
4.3.1 Cylindrical Wiener processes
96(2)
4.3.2 Approximations of stochastic integrals
98(1)
4.3.3 Comments on the Brownian sheet approach
99(2)
4.4 Properties of the stochastic integral
101(4)
4.5 Ito formula
105(4)
4.6 The stochastic Fubini theorem
109(4)
4.7 Remarks on generalization of the integral
113(2)
II Existence and Uniqueness
115(122)
5 Linear equations with additive noise
117(33)
5.1 Basic concepts
117(4)
5.1.1 Concept of solutions
117(2)
5.1.2 Stochastic convolution
119(2)
5.2 Existence and uniqueness of weak solutions
121(6)
5.3 Continuity of weak solutions
127(3)
5.4 Regularity of weak solutions in the analytic case
130(8)
5.4.1 Basic regularity theorems
130(5)
5.4.2 Regularity in the border case
135(3)
5.5 Regularity of weak solutions in the space of continuous functions
138(9)
5.5.1 The case when A is self-adjoint
139(3)
5.5.2 The case of a skew-symmetric generator
142(2)
5.5.3 A perturbation result
144(3)
5.6 Existence of strong solutions
147(3)
6 Linear equations with multiplicative noise
150(30)
6.1 Strong, weak and mild solutions
150(9)
6.1.1 The case when B is bounded
157(2)
6.2 Stochastic convolution for contractions semigroups
159(3)
6.3 Stochastic convolution for analytic semigroups
162(6)
6.3.1 General results
162(4)
6.3.2 Variational case
166(1)
6.3.3 Self-adjoint case
167(1)
6.4 Existence of mild solutions in the analytic case
168(6)
6.4.1 Introduction
168(1)
6.4.2 Existence of solutions in the analytic case
169(5)
6.5 Existence of strong solutions
174(6)
7 Existence and uniqueness for nonlinear equations
180(38)
7.1 Equations with Lipschitz nonlinearities
180(17)
7.1.1 The case of cylindrical Wiener processes
193(4)
7.2 Nonlinear equations on Banach spaces: Additive noise
197(15)
7.2.1 Locally Lipschitz nonlinearities
198(4)
7.2.2 Dissipative nonlinearities
202(4)
7.2.3 Dissipative nonlinearities and general initial conditions
206(3)
7.2.4 Dissipative nonlinearities and general noise
209(3)
7.3 Nonlinear equations on Banach spaces: Multiplicative noise
212(3)
7.4 Strong solutions
215(3)
8 Martingale solutions
218(19)
8.1 Introduction
218(2)
8.2 Representation theorem
220(5)
8.3 Compactness results
225(5)
8.4 Proof of the main theorem
230(7)
III Properties of solutions
237(142)
9 Markov properties and Kolmogorov equations
239(39)
9.1 Regular dependence of solutions on initial data
239(9)
9.1.1 Differentiability with respect to the initial condition
243(3)
9.1.2 Comments on stochastic flows
246(2)
9.2 Markov and strong Markov properties
248(9)
9.2.1 Case of Lipschitz nonlinearities
248(8)
9.2.2 Markov property for equations in Banach spaces
256(1)
9.3 Kolmogorov's equation: Smooth initial functions
257(6)
9.3.1 Bounded generators
258(2)
9.3.2 Arbitrary generators
260(3)
9.4 Kolmogorov's equation: General initial functions
263(12)
9.4.1 Linear case
263(5)
9.4.2 Nonlinear case: mild solutions
268(3)
9.4.3 Nonlinear case: strict solutions
271(4)
9.5 Specific examples
275(3)
10 Absolute continuity and Girsanov's theorem
278(24)
10.1 Absolute continuity for linear systems
278(11)
10.1.1 The case B = B = I
284(5)
10.2 Girsanov's theorem and absolute continuity for non-linear systems
289(8)
10.2.1 Girsanov's Theorem
290(7)
10.3 Application to martingale solutions
297(5)
11 Large time behaviour of solutions
302(44)
11.1 Basic concepts
302(5)
11.2 Linear equations with additive noise
307(10)
11.2.1 Characterization theorem
308(3)
11.2.2 Uniqueness of invariant measure and asymptotic behaviour
311(2)
11.2.3 Strongly Feller case
313(4)
11.3 Linear equations with multiplicative noise
317(10)
11.3.1 Bounded diffusion operators
317(7)
11.3.2 Unbounded diffusion operator
324(3)
11.4 General linear equations
327(2)
11.5 Dissipative systems
329(8)
11.5.1 Regular coefficients
330(1)
11.5.2 Discontinuous coefficients
331(6)
11.6 The compact case
337(9)
11.6.1 Finite trace Wiener processes
338(4)
11.6.2 Cylindrical Wiener processes
342(4)
12 Small noise asymptotic
346(33)
12.1 Large deviation principle
346(13)
12.1.1 Formulation and basic properties
348(3)
12.1.2 LDP for a family of Gaussian measures
351(3)
12.1.3 LDP for linear systems with additive noise
354(3)
12.1.4 LDP for semilinear equations
357(2)
12.2 Exit problems
359(20)
12.2.1 Exit rate estimates
361(6)
12.2.2 Exit place determination
367(6)
12.2.3 Explicit formulae for gradient systems
373(6)
A Linear deterministic equations
379(27)
A.1 Cauchy problems and semigroups
379(2)
A.2 Basic properties of Co-semigroups
381(2)
A.3 The Cauchy problem for non homogeneous equations
383(3)
A.4 The Cauchy problem for analytic semigroups
386(10)
A.4.1 Analytic generators
386(2)
A.4.2 Variational generators
388(1)
A.4.3 Fractional powers and interpolation spaces
389(4)
A.4.4 Regularity of solutions for the homogeneous Cauchy problem
393(1)
A.4.5 Regularity for the non homogeneous problem
394(2)
A.5 Examples of deterministic systems
396(10)
A.5.1 Delay systems
396(1)
A.5.2 Heat equation
397(3)
A.5.3 Heat equation in variational form
400(2)
A.5.4 Wave and plate equations
402(2)
A.5.5 Wave and plate equations with strong damping
404(2)
B Some results on control theory
406(9)
B.1 Controllability and stabilizability
406(1)
B.2 Comparison of images of linear operators
407(3)
B.3 Operators associated with control systems
410(3)
B.3.1 Characterization of Im
410(1)
B.3.2 Characterization of Im L
411(2)
B.4 Controllability of a nonlinear system
413(2)
C Nuclear and Hilbert - Schmidt operators
415(5)
D Dissipative mappings
420(7)
D.1 Subdifferential of the norm
420(3)
D.2 Characterizations of dissipative mappings
423(1)
D.3 Continuous dissipative mappings
424(3)
Bibliography 427(24)
Index 451