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Lasota A., Mackey M.C. — Probabilistic Properties of Deterministic Systems
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Название: Probabilistic Properties of Deterministic SystemsАвторы: Lasota A., Mackey M.C. Аннотация: This book shows how densities arise in simple deterministic systems. Recently there has been explosive growth in interest in physical, biological, and economic systems that can be profitably studied using densities. Due to the inaccessibility of the mathematical literature there has been little diffusion of the applicable mathematics into the study of these 'chaotic' systems. This book will help to bridge that gap. The authors give a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial differential equations. They have drawn examples from many scientific fields to illustrate the utility of the techniques presented. The book assumes a knowledge of advanced calculus and differential equations, but basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed
Язык: Рубрика: Физика /Динамические системы /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1985Количество страниц: 358Добавлена в каталог: 17.09.2005Операции: Положить на полку |
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Koopman operator and motion on torus 190—192 Koopman operator and rotational transformation 69—70 Koopman operator, relation to ergodicity 54 68 69—70 189 190 194 204 Koopman operator, relation to Frobenius — Perron operator 43 177 215—217 Koopman operator, relation to infinitesimal operators 184—185 204 Koopman operator, relation to mixing 68 194 Koopman operator, relation to ordinary differential equations 184—185 Langevin equation 315—317 325—326 332—333 Law of Large Numbers, strong 276—277 Law of large numbers, weak 275 Lebesgue dominated convergence theorem 18 Lebesgue integral 16—18 Lebesgue integral on product spaces 25 Lebesgue integral, relation to Riemann integral 19 Lebesgue measure 27 Lebesgue monotone convergence theorem 18 Left lower semicontinuous function 111 Length of gradient on manifolds 156 Liapunov function 104—105 108—109 284 287—288 329—330 336—337 Linear abstract Boltzmann equation 231 Linear Boltzmann equation 228 231 241 264 Linear subspace 83 Linear Tjon — Wu equation 245 Linearly dense set 28 Liouville equation 203—204 260 329 Liouville's theorem 203 Lorenz equations 124—125 Lower semicontinuous function 111 Lower-bound function 96 Lower-bound function, conditions for existence 111—113 159 Lower-bound function, relation to asymptotic stability 96 102 175 Manifold 151—158 Manifold, connected 156 Manifold, d-dimensional 152 Manifold, geodesic flow on 199 Markov operator 32 Markov operator, adjoint operator to 44 Markov operator, and Frobenius — Perron operator 38 Markov operator, and linear abstract Boltzmann equation 231 232 236 238 Markov operator, and parabolic equation 326 Markov operator, and stochastic perturbation 279 284 Markov operator, asymptotic periodicity 86—90 Markov operator, asymptotic stability 95 Markov operator, constrictive 87—88 Markov operator, contractive property of 34 175 Markov operator, ergodic 72 92 Markov operator, exact 72 93 Markov operator, fixed point of 35 Markov operator, lower-bound function for 96 Markov operator, mixing 72 94 Markov operator, properties of 33 Markov operator, relation to entropy 254—257 Markov operator, semigroup of 174 Markov operator, stability property of 34 175 Markov operator, stationary density of 36 Markov operator, strongly constrictive 87—88 Markov operator, weak continuity of 43 Markov operator, weakly constrictive 87 Markov operator, with stochastic kernel 101 217 241 Mathematical expectation 269 Maximal entropy 250—253 Maxwellian distribution 336 Mean value of function 114 Mean value of random variable 269 Measurable function 15 Measurable function, space of 21 Measurable set 14 Measurable transformation 36 Measure 14 Measure space 14 Measure space, 14 Measure space, finite 15 Measure space, normalized 15 Measure space, probabilistic 15 Measure space, product of 24 Measure, absolutely continuous 36 Measure, Borel 14 Measure, complete 27 Measure, density of 36 Measure, invariant 45 169 Measure, Lebesgue 27 Measure, Wiener 298—299 Measure-preserving transformation 45 169—170 Metric, Riemannian 155 Mixing of Anosov diffeomorphism 71 Mixing of baker transformation 60 Mixing of dyadic transformation 61 Mixing, dynamical system 172 Mixing, illustrated 64 Mixing, Markov operator 72 94 Mixing, necessary and sufficient conditions for via Frobenius — Perron operator 65 66 194 Mixing, necessary and sufficient conditions for via Koopman operator 68 194 Mixing, relation to ergodicity, exactness, and K-automorphisms 62 73 76 173 Mixing, semidynamical system 172 Mixing, transformation 59 Modulo zero equality 34 Moments of solutions 325—326 Nonanticipative 305 Nonsingular semidynamical system 173 Nonsingular transformation 36 Nontrivial lower-bound function 96 Norm in 21 Norm of vector on manifold 156 Normalized measure space 15 Normalized Wiener process 294 One-dimensional Brownian motion 294 One-dimensional Wiener process 294 Operator, constrictive 87—88 Operator, contractive 34 175 Operator, Frobenius — Perron 37 Operator, infinitesimal 180 Operator, Koopman 42—44 Operator, Markov 32 Operator, resolvent 201 oscillators 136—138 192 Parabolic equation 326 Parabolic transformation see "Quadratic transformation" Parabolicity condition 323 Paradox of weak repellor 11 126 Partition function 253 Phase space 1 163 166 Phillip's perturbation theorem 210 Piece wise convex transformations 128 Piecewise monotonic mappings 119 123 147 Poincare map 220 Poincare recurrence theorem 59 Poisson bracket 187 Poisson processes 222—226 Probabilistic measure space 15 220 Product measure 24 227 Product space 24 227 Proper cylinder 195 Quadratic transformation 1 7 47 50 142 255 r-adic transformation 9 46—47 70 Radon — Nikodym theorem 20 23 Random number generator 147 Random variable 221 Random variable, density of 221 Random variable, independent 221 266 Random variable, mathematical expectation of 269 Random variable, mean value of 269 Random variable, standard deviation of 272 Random variable, variance of 271 Randomly applied stochastic perturbation 277—282 Regular Ito sum 305 Renyi transformation 119 Resolvent operator 201 Riemann integral, relation to Lebesgue integral 19 Riemannian metric 155 Right lower derivative 111 Rock drilling 138 Rotation on circle 56 69 171 Rotation on torus 190 Sample path 222 Scalar product 23 Scalar product on manifolds 155 Semidynamical system 168 Semigroup of contracting operators 177—178 Semigroup of contractions 178 Semigroup of Frobenius — Perron operator 173—174 Semigroup of Koopman operator 177 Semigroup of transformations 169 Simple function 17 Space and time averages 58 170 Space of measurable functions 21 Space, adjoint 22 Spectral decomposition theorem 88 Sphere bundle 199 Stability property of Markov operators 34 Standard deviation 272 State space 1 Stationary density 36 Stationary independent increments 222 Statistical stability 95 Statistical stability, relation to asymptotic stability 95 Statistical stability, relation to exactness 100—101 Statistically stable transformation, construction of 143 Stirling's formula 235 Stochastic convergence 274 Stochastic differential equations 293 313 Stochastic differential equations, relation to Fokker — Planck equation 317—318 Stochastic integrals 305 311 Stochastic kernel 101 217 241 Stochastic perturbation, constantly applied 282 Stochastic perturbation, randomly applied 277 Stochastic perturbation, small 277 289 Stochastic processes 221 Stochastic processes with independent increments 222 Stochastic processes with stationary independent increments 222 Stochastic processes, continuous time 222 Stochastic processes, discrete time 222 Stochastic semigroup 174 Stochastic semigroup, relation to Fokker — Planck equation 327 Stratonovich sum 308 Strong convergence 27 Strong convergence, Cauchy condition for 30 Strong Law of Large Numbers 276 Strong precompactness 78 Strong precompactness, conditions for 79 Strong repellor 128—131 Strongly constrictive Markov operator 87 Support 34 Support and Frobenius — Perron operator 39 Support, compact 181 Tangent space 153 Tangent vector 153 Tent map 142 Time and space averages 58 170 Torus 162 Torus, Anosov diffeomorphism on 52 Torus, d-dimensional 190 Torus, exact transformation on 162 Torus, rotation on 191 Trace of dynamical system 167 Trajectory 166 Trajectory, versus density 10 Transformation, asymptotically periodic 131 Transformation, convex 128 Transformation, ergodic 53 171 Transformation, exact 62 173 Transformation, factor of 75 Transformation, Frobenius — Perron operator for 6 37 173 Transformation, Koopman operator for 42 177 Transformation, measurable 36 Transformation, measure-preserving 45 Transformation, mixing 59 172 Transformation, nonsingular 36 Transformation, piecewise monotonic 119 123 147 Transformation, statistically stable 95 100—101 Transformation, weakly mixing 73 Triangle inequality 22 Trivial 74 Trivial set 53 171 Uniform parabolicity 323 Unit volume function 157 Variance of function 114 Variance of random variable 271 Variance of Wiener process 295 Variation of function 115 Vector, norm of 156 Vector, Scalar product of 155 Vector, space 22 Von Neumann series 233 Weak Cesaro convergence 27 Weak continuity 43 Weak convergence 27 Weak law of large numbers 275 Weak precompactness 78 Weak repellor, paradox of 11 125 Weakly constrictive Markov operator 87 Weakly mixing transformation 73 Wiener measure 298—299 Wiener process, d-dimensional 303 Wiener process, normalized 294—295 Wiener process, one-dimensional 294 Wiener process, variance of 295 Yorke inequality 118
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