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Lasota A., Mackey M.C. Ч Probabilistic Properties of Deterministic Systems
Lasota A., Mackey M.C. Ч Probabilistic Properties of Deterministic Systems

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Ќазвание: Probabilistic Properties of Deterministic Systems

јвторы: Lasota A., Mackey M.C.


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: en

–убрика: ‘изика/ƒинамические системы/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1985

 оличество страниц: 358

ƒобавлена в каталог: 17.09.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
$L^{p}$ distance      22
$L^{p}$ norm      21
$L^{p}$ space      21
$L^{p}$ space, complete      30
$L^{p}$, space adjoint to      22
$\sigma$-algebra      13
$\sigma$-algebra, Borel      14
$\sigma$-algebra, independent      302
$\sigma$-algebra, nonanticipative      305
$\sigma$-algebra, of Borel sets      14
$\sigma$-algebra, trivial      74
$\sigma$-finite measure space      14
Abel inequality      117
Abstract ergodic theorem      81
Acoustic feedback system      139
Adjoint operator      43 44
Almost everywhere (a.e.)      15 33
Almost sure convergence      275
Anosov diffeomorphism      51 52 71Ч72
Arcs, equivalent      153
Asymptotic periodicity      86Ч90 107 131
Asymptotic stability      95 176
Asymptotic stability of Chandrasekhar Ч Muench equation      344
Asymptotic stability of convex transformation      129
Asymptotic stability of expanding mappings on manifolds      160
Asymptotic stability of fluid flow      131
Asymptotic stability of integral operators      102 105
Asymptotic stability of integro-differential equations      337Ч344
Asymptotic stability of Lorenz equations      124
Asymptotic stability of monotonically increasing transformation      119 123
Asymptotic stability of quadratic transformation      142
Asymptotic stability of Renyi transformation      119
Asymptotic stability of stochastically perturbed systems      280 287
Asymptotic stability of strong repellor      129
Asymptotic stability of transformations on R      148
Asymptotic stability, proof via lower-bound function      96 175
Asymptotic stability, relation to conditional entropy      264
Asymptotic stability, relation to exactness      100
Asymptotic stability, relation to Liapunov function      105 329Ч330 336Ч337
Asymptotic stability, relation to statistical stability      95
Asymptotic stability, via change of variables      141
Automorphism      74
Baker transformation      48Ч50 60Ч61 74Ч75 260
Baker transformation, relation to dyadic transformation      50 260
Birkhoff ergodic theorem, continuous time      170
Birkhoff ergodic theorem, discrete time      57
Boltzmann equation and entropy      260
Borel $\sigma$-algebra      14
Borel measure      14 25Ч26
Borel measure on manifolds      157
Borel sets      14
Bounded variation, function of      114Ч119
Brownian motion, d-dimensional      303
Brownian motion, one-dimensional      294
Cartesian product      24
Cauchy condition for convergence      30
Cauchy problem for Fokker Ч Planck equation      322
Cauchy Ч Hoelder inequality      23
Cell proliferation      301Ч302
Cesaro convergence      27
Cesaro convergence of Frobenius Ч Perron operator      65Ч66
Cesaro convergence of Koopman operator      68
Chandrasekhar Ч Muench equation      214 341 344
Change of variables and asymptotic stability      141Ч147
Change of variables in Lebesgue integral      40Ч41
Characteristic function      5
Chebyshev inequality      104 273
Chebyshev polynomials      145
Classical solution of Fokker Ч Planck equation      323
Closed linear subspace      83
Compact support      181
Comparison series      31
Complete measure      27
Complete space      30
Conditional entropy      255Ч256 263Ч264
Connected manifold      156
Constant of motion      187
Constrictive Markov operator      87Ч88
Continuous semigroup      169
Continuous semigroup and ordinary differential equations      187Ч188
Continuous semigroup of contractions      178
Continuous semigroup of contractions and infinitesimal operator      200
Continuous stochastic processes      294
Continuous stochastic semigroup      175
Continuous time stochastic process      221Ч222
Continuous time system and discrete time systems      172 219Ч220
Continuous time system, ergodic      171 172
Continuous time system, exact      173 297Ч302
Continuous time system, mixing      172 194Ч200
Contracting operator      34 177
Convergence, almost sure      275
Convergence, Cauchy condition for      30
Convergence, Cesaro      27
Convergence, comparison series for      31
Convergence, in different $L^{p}$ spaces      29
Convergence, in mean      274
Convergence, stochastic      274
Convergence, strong      27
Convergence, to set of functions      87
Convergence, weak      27
Convergence, weak Cesaro      27
Convex transformation      128Ч130
Counterimage      5
Counting measure      14
Counting process      222
Curvature      199Ч200
CYCLE      92
Cyclical permutation      92
Cylinder      195 298
Dense subset of densities      99
density      5 10 36
Density of absolutely continuous measure      36
Density of random variable      221
Density, evolution of by Frobenius Ч Perron operator      38 215Ч217
Derivative, right lower      111
Derivative, strong      181
Determinant of differential on manifold      158
Diffeomorphism      52
Differential delay equation as semidynamical system      164Ч165
Differential equation as dynamical system      164
Differential of transformation on manifold      154
Differential, determinant of, on manifold      158
Discrete time stochastic process      221Ч222
Discrete time system and Poisson processes      226Ч229
Discrete time system as sampled semidynamical system      219Ч220
Discrete time system, embedded in continuous time system      220
Distance, between function and set of functions      86Ч87
Distance, in $L^{p}$ spaces      22
Distance, on manifold      156
Dyadic transformation      9 61Ч62 260
Dyadic transformation, related to baker transformation      50
Dynamical system      165
Dynamical system and exactness      173
Dynamical system, ergodic      171
Dynamical system, mixing      172
Elementary events      220
Endomorphism      74
entropy      249
Entropy and exact transformations      258 260 262
Entropy and Frobenius Ч Perron operator      257Ч260 262
Entropy and Hamiltonian systems      257Ч258
Entropy and invertible transformations      257
Entropy and Liouville equation      260
Entropy and Markov operators      254Ч257
Entropy of reversible and irreversible systems      260
Entropy, conditional      255
Equivalent arcs on manifold      153
Ergodic Birkhoff theorem      57 170
Ergodic dynamical system      171
Ergodic Markov operator      72 92
Ergodic semidynamical system      171
Ergodic transformation      53
Ergodicity and Hamiltonian systems      205
Ergodicity and linear Boltzmann equation      240Ч241
Ergodicity and rotational transformation      56Ч57 69Ч70 171Ч172
Ergodicity of motion on torus      190Ч192
Ergodicity, conditions for via Frobenius Ч Perron operator      55 65Ч66 86 194
Ergodicity, conditions for via Koopman operator      54 68 189 194 204
Ergodicity, illustrated      63
Ergodicity, necessary and sufficient conditions for      53
Ergodicity, relation to mixing, exactness, and K-automorphisms      73
Essential supremum      23
Euler Ч Bernstein equations      315
Events, elementary      220
Events, independent      221
Events, mutually disjoint      221
Exact Markov operator      72 93
Exact semidynamical system      173
Exact semidynamical system with continuous time      297Ч302
Exact semigroup of linear Boltzmann equation      241
Exact transformation      62
Exactness and entropy      258 262
Exactness, illustrated      65
Exactness, necessary and sufficient conditions for via Frobenius Ч Perron operator      65Ч66 194
Exactness, of r-adic transformation      70
Exactness, of transformations on torus      162
Exactness, relation to dynamical systems      173
Exactness, relation to ergodicity, mixing, and K-automorphisms      62 73 76
Exactness, relation to statistical stability      100Ч101 143
Expanding mappings      158Ч162
Factor of transformation      75
Finite measure space      15
First return map      219
Fixed point of Markov operator      35
Fluid flow      131
Fokker Ч Planck equation, and Cauchy problem      322
Fokker Ч Planck equation, and Liouville equation      329
Fokker Ч Planck equation, and stochastic semigroups      327
Fokker Ч Planck equation, asymptotic stability of solutions      330
Fokker Ч Planck equation, classical solution      323
Fokker Ч Planck equation, derivation of      318Ч322
Fokker Ч Planck equation, existence and uniqueness of solutions      323Ч324
Fokker Ч Planck equation, for Langevin equation      325 332
Fokker Ч Planck equation, for second-order system      335
Fokker Ч Planck equation, for stochastic differential equations      318
Fokker Ч Planck equation, fundamental solution      323
Fokker Ч Planck equation, generalized solution      327
Fokker Ч Planck equation, stationary solutions      332 333 334 336
Forced oscillator      136Ч138
Frobenius Ч Perron operator      37 173Ч174
Frobenius Ч Perron operator and evolution of densities      4Ч12 38
Frobenius Ч Perron operator and invariant measure      46 189 203
Frobenius Ч Perron operator and Koopman operator      177
Frobenius Ч Perron operator and Liouville equation      203
Frobenius Ч Perron operator and ordinary differential equations      185Ч187
Frobenius Ч Perron operator and semidynamical systems      173Ч174 189
Frobenius Ч Perron operator as Markov operator      38
Frobenius Ч Perron operator for Anosov diffeomorphism      51Ч52
Frobenius Ч Perron operator for baker transformation      48Ч50
Frobenius Ч Perron operator for dyadic transformation      9
Frobenius Ч Perron operator for Hamiltonian system      187Ч188
Frobenius Ч Perron operator for invertible transformations      42
Frobenius Ч Perron operator for quadratic map      7 47
Frobenius Ч Perron operator for r-adic transformation      9 46
Frobenius Ч Perron operator for transformations on $R^{2}$      40
Frobenius Ч Perron operator for transformations on R      10 38 148
Frobenius Ч Perron operator, relation to entropy      257Ч260 262
Frobenius Ч Perron operator, relation to ergodicity      55 65Ч66 86 194
Frobenius Ч Perron operator, relation to infinitesimal operator      185Ч187
Frobenius Ч Perron operator, relation to Koopman operator      43 177 215Ч217
Frobenius Ч Perron operator, relation to mixing      65Ч66 194
Frobenius Ч Perron operator, semigroups of      173Ч174
Frobenius Ч Perron operator, support of      39
Frobenius Ч Perron operator, weak continuity of      43Ч44
Fubini's theorem      25
Function of bounded variation      115
Function, left lower semicontinuous      111
Function, lower semicontinuous      111
Function, support of      34
Fundamental solution of Fokker Ч Planck equation      323Ч324
Gas dynamics      194Ч198 244Ч247
Gaussian density      251 286 294 303
Gaussian kernel      176 208 214 218 324
Generalized solution of Fokker Ч Planck equation      327
Geodesic      199
Geodesic, flow      199
Geodesic, motion on      198Ч200
Gibbs canonical distribution function      253
Gibbs inequality      249
Gradient of function      152
Gradient, length of      156
Hahn Ч Banach theorem      83
Hamiltonian      187
Hamiltonian, system      187 192 198 204 205 257
Hat map      142
Hausdorff space      152
Heat equation      176 208
Henon map      50
Hille Ч Yosida theorem      200
Homeomorphism      152
Ideal gas      194Ч198 244Ч247
Independent $\sigma$-algebras      302
Independent events      221
Independent increments      222
Independent random variables      221 266
Indicator function      5
Inequality, Cauchy Ч Hoelder      23
Inequality, triangle      22
Infinitesimal operator      180
Infinitesimal operator and differential equations      181
Infinitesimal operator and ergodicity      189
Infinitesimal operator and Frobenius Ч Perron operator      185Ч187 203
Infinitesimal operator and Hamiltonian systems      187
Infinitesimal operator and Hille Ч Yosida theorem      200
Infinitesimal operator and invariant measure      203
Infinitesimal operator and Koopman operator      184Ч185 204
Infinitesimal operator and ordinary differential equations      183Ч185
Infinitesimal operator and partial differential equation      182Ч183
Infinitesimal operator as strong derivative      181
Infinitesimal operator of continuous semigroup of contractions      200
Infinitesimal operator, illustrated by heat equation      208
Infinitesimal operator, illustrated by parabolic differential equations      208Ч209
Integrable function      17
Integral, Ito      304Ч309
Integral, Lebesgue      16Ч18
Integral, Stochastic      305 311
Integro-differential equations      213 214 337 341
Intermittency      131
Invariant measure      45 169
Invariant measure and differential equations      203
Invariant measure and Frobenius Ч Perron operator      46 189
Invariant measure and Hamiltonian systems      204
Invariant measure and infinitesimal operators      203
Invariant measure and Liouville's theorem      203Ч204
Invariant measure for monotonic transformations      132
Invariant set      53 171
Invertibility      50 62 165 257 260
Ito integral      304Ч309
Ito sum      305
Jacobian matrix      41
Jensen inequality      253
Joint density function      267
K-automorphism      73Ч76
K-automorphism and exactness      76
K-automorphism and geodesic flows      200
K-automorphism and mixing      76
Kolmogorov automorphism      73
Kolmogorov equation      see "Fokker Ч Planck equation"
Koopman operator      42Ч44 177
Koopman operator and Anosov diffeomorphism      71Ч72
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