References

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4 288 References 76. M. Vishik and A. Fursikov. Mathematical Problems of Statistical Hydrodynamics, Kluwer, London (1988). 77. M. Yor. Existence et unicité de diffusions à valeurs dans un espace de Hilbert, Ann. Inst. H. Poincaré B 10, (1974). 78. K. Yosida. Functional Analysis, Springer, New York (1980). 79. J. Zabczyk. Linear stochastic systems in Hilbert spaces; spectral properties and limit behaviour, Banach Cent. Publ. 41, (1985).

5 Index A Abstract Cauchy problem, 4, 11 B Burkholder inequality, 87 C Chapman Kolmogorov equation, 256 Covariance of a Gaussian random variable, 18 of Gaussian stochastic convolution, 259 Cylindrical Wiener process, 19 D Datko s theorem stochastic analogue of, 230 Dissipative system, 237 Dissipativity condition, 237 Doob s Maximal Inequalities, 22 E Eigenvalues, 18 Eigenvectors, 18 Energy equality, 208 F Factorization formula, 89 Factorization technique, 87 Feller property, 110 G Gaussian measure with covariance Q,19 Gaussian random variable, 17 cylindrical standard, 17 with covariance Q,18 Gaussian semigroup, 4 Gibbs measure, 194 H Heat equation, 3 in R d,13 one dimensional, 3 stochastic, 248 Hille Yosida theorem, 8 I Increasing process, 22 of Q Wiener process, 23 Infinitesimal generator, 6 Invariant probability measure, 256 the linear case, 260 Itô s formula the case of cylindrical Wiener process, 69 the case of Q Wiener process, 61 the variational case, 226 K Kolmogorov s backward equation, 118 L Lebesgue s DCT generalized, 100 Lévy s theorem, 51 Linear equation with additive noise, 259 Lions s lemma, 15 Lions s theorem, 15 Lyapunov function, 206, 216 M Markov process, 107 property, 107 transition function, 256 transition probabilities, 255 L. Gawarecki, V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions, Probability and Its Applications, DOI / , Springer-Verlag Berlin Heidelberg

6 290 Index Martingale, 21 square integrable, 22 Martingale Representation Theorem I, 51 Martingale Representation Theorem II, 53 Maxwell probability measure, 260 Monotonicity, 173 weak, 173 N Navier-Stokes equation, 258 Norm graph, 7 Hilbert Schmidt, 24 trace, 18 O Operator adjoint, 5 coercive, 246 Hilbert Schmidt, 24 non-negative definite, 5 symmetric, 5 trace class, 18 P Poincaré inequality, 283 Probability space, 17 complete, 17 filtered, 19 Process, 19 elementary, 25 bounded, 25 Feller, 110 Gaussian, 19 Ornstein Uhlenbeck, 259 progressively measurable, 38 weakly positive recurrent, 278 Prokhorov s theorem, 148 Q Q Wiener process, 20 continuity, 20 properties, 21 Quadratic variation process, 22 existence and uniqueness, 22 of Q Wiener process, 23 R Reaction Diffusion Equation, 282 Recurrent region, 274 Resolvent, 8 set, 8 S Semigroup, 5 adjoint, 7 C 0,5 compact, 6 differentiable, 6 exponentially stable, 260 Feller, 110 of contractions, 6 pseudo contraction, 6 semigroup property, 5 strong continuity property, 5 strongly continuous, 5 uniformly bounded, 6 uniformly continuous, 6 Sequence stochastically bounded, 129 Sobolev space, 11 Solution, 74 classical, 11 martingale, 75 mild, 12, 75 mild integral, 74 strong, 74 weak, 75 weakened, 74 Spin system, 193 lattice, 194 Stability, 203 deterministic condition for, 204 exponential in the m.s.s., 213 of the mild solution, 213 of the variational solution, 227 exponential stability of the solution of the Cauchy problem, 203 in probability of the zero solution, 223 of a solution to a linear equation with additive noise, 259 Stochastic convolution, 76 Gaussian, 259 Stochastic differential equation of evolution type, 247 semilinear, 73 variational, 152 Stochastic Fubini Theorem, 57 Stochastic integral with respect to a Q Wiener process of a process in P(K Q,H),42 of a process in 2 (K Q,H),34 with respect to a standard cylindrical Wiener process, 45 Stochastic process, 260 ultimately bounded in the m.s.s., 260 weakly recurrent, 274

7 Index 291 Strong Markov process, 183 Strong Markov property, 183 T Theorem about convolution, 94 Tight family of measures, 148 Trace, 18 U Uniqueness, 173 pathwise, 173 Usual conditions, 19 V Variational method, 15 Variational solution, 152 strong, 152 weak, 152 Y Yosida approximation, 9