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Guiseppe Da Prato — Stochastic equations in infinite dimensions
Guiseppe Da Prato — Stochastic equations in infinite dimensions

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Название: Stochastic equations in infinite dimensions

Автор: Guiseppe Da Prato

Аннотация:

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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 476

Добавлена в каталог: 18.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C_{0}$ semigroup of linear operators      380
$m$th - Wick power      7
$Q$ - Wiener process      86
$Q$ - Wiener process with respect to filtration      90
$\alpha$ - H$\ddot{o}$lder continuous process      71
$\pi$-system      16
$\sigma$-field      15
Absolute continuity      278
Absolutely continuous measure      59 282 295
Adapted process      75
Additive noise      xvii 117 197 307
Analytic case      169
Analytic semigroup      130 387
Approximately controllable system      406
Associated control system      358 375
Birkhoff theorem      306
Bochner integrable      19
Bochner theorem      49
Border case      135
Borel $\sigma$-field      15
Brownian Bridge process      9
Brownian sheet      100
Cable equation      10
Cameron - Martin formula      68
Cauchy problem      379
Cauchy problem, parabolic      387
Chapman - Kolmogorov equation      251
Characteristic function      35 36
Closed operator      381
Compact case      337
Compact family of measures      32
Complete probability space      18
Conditional expectation      27
Continuous dissipative mapping      425
Continuous with probability      1 71
Contraction $C_{0}$ semigroup      382
Control theory      xv
Correlation operator      26
Covariance of a measure      54
Covariance operator      11 26
Cross quadratic variation      80 81
Cylindrical set      35
Cylindrical Wiener      126
Cylindrical Wiener process      96 342
Delay system      123 396
Dependence on initial data      239
Deterministic Cauchy problem      118
Dirichlet boundary conditions      398
Dirichlet realisation      401
Dissipative mapping      423
Dissipative nonlinearities      202 206 209 331
Dissipativity condition      330
Distribution of a random variable      18
Dual semigroup $P^{*}_{t}$      303
Elementary process      90
Equation of population genetics      10
Equation of stochastic quantization      6
Equation of the free field      6
Equation with Lipschitz nonlinearities      180
Equations on Banach spaces      197
Equations with mulitplicative noise      317
Equivalent measures      59
Evolutionary Liapunov equation      319
Existence and uniqueness results xiv existence of a continuous modification      192
Existence of a martingale solution      219
Existence of strong solutions      147
Exit place      367
Exit problem      xvii
Exit rate      361
Exit set      367
Exponential estimates of Freidlin - Wentsell      350
Exponentially stable generator      407
Exponentially stable semigroup      356
Factorization formula      229
Factorization method      xvi 128 184 219
Factorization procedure      219
Feldman - Hajek theorem      58 63 279
Feller property      249
Fernique theorem      xv 36 37
Feynman - Kac formula      262
Filtering equation xiii filtration      75
First minimum energy problem      373
Fractional power and interpolation space      389
Fractional powers      389
Gaussian distribution      36
Gaussian measure      37
Gaussian measure in Banach spaces      36
Gaussian measure on Hilbert spaces      53
Gaussian processes      9 83
Generalization of the stochastic integral      113
Generalized solution      206 208
Generated $\sigma$-field      16
Girsanov theorem      xvii 290
Gradient systems      374
Graph norm      381
Growth conditions      332
Heat equation      124 397
Hellinger integral      58
Hilbert - Schmidt operator      418
Increasing process      80
Independent $\sigma$-fields      28
Independent random variables      28
Infinitesimal generator      xv 380
Initial value problem      379
Interpolation sapces      390
Invariant measure      303
It$\hat{o}$ formula      105
Kolmogorov's backward equation      257
Kolomorov test      74 142
Large deviation principle      348
Law of random variable      18
Levy's theorem      80 291
Levy's theorem, generalization      89
Liapunov equation      318 407
Lifts of diffusion      1
Lipschitz nonlinearities      198
Local inversion theorem      243
Localization procedure      94
Lower exit rate      362
Markov processes      82 248
Markov property xiv      256
Martingale      77
Martingale property of the stochastic integral      96
Martingale solutions      xvi 219 297
Maximal inequalities for real-valued submartingales      78
Maximal regularity      162 395
Mean of a measure      54
Mean square continuous process      71
Mean square stability      xvii 318
Measurable decomposition      25
Measurable mapping      15
Measurable process      70
Measurable selectors      26
Measurable space      15
Mild solution      152 168 182
Mild solution of Kolmogorov's equation      268
Mild solution of the Liapunov equation      319
Minimal energy      362 413
Mixed conditions      399
Modification of a process      70
Motion of a string      4
Multiplicative noise      xvii 150 212
Nemitskii operator      200 336 424
Neumann problem      200
Neumann realization      401
Non-degenerate processes      314
Normal filtration      75
Nuclear operator      26 415
Null controllable system      264 406
Operator-valued random variables      23
Ornstein-Uhlenbeck process      xiii
Parabolic type equations      xiii
Plate equation      402 405
Positive definite      49
Predictable $\sigma$-field      76 93
Predictable process      76
Predictable sets      76
Predictable version      153
Probability measure      18
Probability measures in Hilbert space      48
Probability space      18
Process, integrable      77
Product $\sigma$-field      20
Progressively measurable      75
Prokhorov theorem      32
Pseudo-contraction $C_{0}$ semigroup      382
Pseudo-inverse      407
Quadratic variation      81
Quasi-potential      373
Random variables      15
Rate function      9 348 349
Rate of convergence      348
Rate of divergence      361
Reaction-difusion equation      9
Regular measure      31
Regularity of solutions      393
Regularity of the stochastic convolution      146
Regularity of weak solutions      138
Relatively compact family of measures      32
Renormalization      8
Representation theorem      220
Reproducing kernel space      40 41 351
Resolvent operator      381
Resolvent set      381
Second minimum energy problem      373 378
Self-adjoint case      167
Semigroup theory      xv
Semigroup, exponentially stable      406
Semigroup, strongly Feller at a moment $r > 0$      315
Semilinear equations      xv 10 357
Simple random variable      15
Simple stopping time      82
Singular measures      59
Skew-symmetric generator      142
Skorohod embedding theorem      33 219
Sobolev embedding theorem      399
Spatial regularity      131
Spectrum      381
Square integrable martingales      79
State reachable      406
Stationary      83
Stationary measure      303
Stochastic calculus xiv stochastic characteristics      175
Stochastic convolution      119 153 159 162
Stochastic delay equations xiv      2
Stochastic evolution equations xiii stochastic flows xiii      2 247 248
Stochastic Fubini theorem      xvi 109 184
Stochastic integral      90
Stochastic process      70
Stochastic quantization equations      7
Stochastic wave equation      158
Stochastically continuous process      70
Stochastically integrable process      94
Stopping time      82
Strict solution      260
Strict solutions of Kolomogorov's equation      258 271
Strong Bochner integral      24
Strong Markov processes      254
Strong solution      118 152 174 182 215
Strongly damped wave equation      172 404
Strongly Feller      249 313
Strongly measurable      24
Subdifferential      198 420
Submartingale      77
Supermartingale      77
Symmetric and Gaussian measure      120
Symmetric Gaussian distribution      37
System, approximately controllable      410
System, means square stable      327
System, null controllable      410
System, null controllable in time r      407
System, stabilizable      407
Temporal regularity      131
Tight family      32 226
Trace of $I$      416
Trajectories      70
Transition function      249
Transition semigroups      xv
Type of the semigroup S($\cdot$)      382
Uniformly attracted set      359
Upper exit rate      362
Variation of constants formula      383
Variational case      166 173 174 324
Variational generators      388
Variational operator      388
Version of a process      70
Wave equation      125 402
Weak solution      119 121 127 130 152 182 219 383
White noise      9 43 48 36
Wiener processes      xiii
Yosida approximations      129 382 425
Zakai equation      3 178
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