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Prevot C., Rockner M. — Concise Course on Stochastic Partial Differential Equations
Prevot C., Rockner M. — Concise Course on Stochastic Partial Differential Equations

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Название: Concise Course on Stochastic Partial Differential Equations

Авторы: Prevot C., Rockner M.


These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. All kinds of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. To keep the technicalities minimal we confine ourselves to the case where the noise term is given by a stochastic integral w.r.t. a cylindrical Wiener process.But all results can be easily generalized to SPDE with more general noises such as, for instance, stochastic integral w.r.t. a continuous local martingale.

There are basically three approaches to analyze SPDE: the "martingale measure approach", the "mild solution approach" and the "variational approach". The purpose of these notes is to give a concise and as self-contained as possible an introduction to the "variational approach". A large part of necessary background material, such as definitions and results from the theory of Hilbert spaces, are included in appendices

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 144

Добавлена в каталог: 19.06.2008

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Предметный указатель
Bochner inequality      107
Bochner integral      105
Boundedness      56
Burkholder — Davis inequality      119
Coercivity      56
Conditional expectation      17
Covariance operator      6
Demicontinuity      56
Elementary process      22
Existence of a unique solution in finite dimensions      44
Fundamental theorem of calculus      108
Gaussian law      6
Gaussian measure      5
Gaussian random variable      9
Gaussian, representation and existence of a      9
Gelfand triple      55
Hemicontinuity      56
Hilbert — Schmidt norm      111
Ito isometry      25
Local weak monotonicity      44
Localization lemma      45
Localization procedure      30
Martingale      19
Maximal inequality      20
Normal filtration      16
Operator, Hilbert — Schmidt      110
Operator, nonnegative      109
Operator, nuclear      109
Operator, symmetric      109
Operator, trace class      109
p-Laplacian      65
Predictable $\sigma$-field      27
Predictable process      28
Pseudo inverse      115
Quadratic variation of the stochastic integral      37
Sobolev space      62
Solution, analytically strong      133
Solution, analytically weak      134
Solution, infinite dimensions      44
Solution, mild      133
Solution, strong      123
Solution, weak      121
Square root of a linear operator      25
Stochastic basis      121
Stochastic differential equation, Burgers equation      3
Stochastic differential equation, heat equation      65
Stochastic differential equation, in finite dimensions      44
Stochastic differential equation, in infinite dimensions      55
Stochastic differential equation, Navier — Stokes equation      3
Stochastic differential equation, porous media equation      3
Stochastic differential equation, reaction diffusion equation      3
Stochastic integral, w.r.t. a standard Wiener process      22
Stochastically integrable process      30
Strong measurability      106
Trace      109
Uniqueness, pathwise      122
Uniqueness, strong      123
Uniqueness, weak      122
Weak monotonicity      56
Wiener process as a martingale      21
Wiener process with respect to a filtration      16
Wiener process, standard      13
Wiener process, standard, representation and existence of a      13
Yamada — Watanabe, Theorem of      124
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