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Port S.C., Stone C.J. — Brownian motion and classical potential theory
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Название: Brownian motion and classical potential theory
Авторы: Port S.C., Stone C.J.
Аннотация: The importani and beautiful connection between Brownian motion and classical potential theory was first investigated by Kakutani, Kac, and Doob. Hunt then showed that potential theory could be developed by probabilistic methods for a large class of transient Markov processes. There is now a large literature on potential theory for transient Markov processes, including books by Dynkin, ltd and McKean, Meyer, and Blumenthal and Getoor. Recurrent potential theory was first developed for random walks by Spitzer and for Markov chains by Kemeny and Snell. Further work in this area has been done by a number of people, including the present authors Port and Stone and Stone(see books by Spitzer, Kemeny, Snell and Knapp, and Revuz).
In spite of the large amount of written material on probabilistic potential theory, the connection between Brownian motion and classical potential theory remains largely inaccessible to nonexperts. The purpose of this book is to give an account of this connection, accessible to readers familiar with real variable theory and probability theory as usually developed at the graduate level (a knowledge of Chung should suffice). No prior knowledge of either Brownian motion or classical potential theory is assumed...
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Рубрика: Физика /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 1978
Количество страниц: 246
Добавлена в каталог: 30.10.2010
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Предметный указатель
Almost everywhere 17
Averaging property 85
Balayage problem, Green potential 172
Balayage problem, logarithmic potential 212
Barrier 140
Blumenthal's zero-one law 8
Borel-Cantelli lemma, extended 65
Boundary value 87
Capacitability theorem 11 204 227
Capacity, 42
Capacity, Green 190
Capacity, Green, inner 203
Capacity, Green, outer 203
Capacity, logarithmic 80
Capacity, logarithmic, inner 226
Capacity, logarithmic, outer 226
Capacity, Newtonian 57
Cauchy density 29
Cauchy sequence 183
Cauchy sequence, weak 183
Charge 44
Complete maximum principle 210
Condenser measure, Green potential 194
Condenser measure, logarithmic potential 219
Condenser theorem, Green potential 195
Condenser theorem, logarithmic potential 219
Cone 26
Cone, condition for recurrence 26
Cone, condition for regularity 30
Continuity principle, Green potential 163
Continuity principle, logarithmic potential 207
Contraction mapping 223
Convergence, complete 18
Convergence, strong 183
Convergence, vague 18
Convergence, weak 183
Dirichlet principle 88
Dirichlet problem 88
Dirichlet problem, exterior 90
Dirichlet problem, generalized 90
Dirichlet problem, stochastic 154
Dirichlet space 195
Domination principle 175
Eigenfunction 122
Eigenvalue 122
Energy principle, Green potential 181
Energy principle, logarithmic potential 214
Energy, Green potential 180 182
Energy, Green potential, mutual 181
Energy, logarithmic potential 213
Equilibrium charge distribution 204
Equilibrium measure, -potential 42
Equilibrium measure, Green potential 190
Equilibrium measure, linear potential 84
Equilibrium measure, logarithmic potential 76
Equilibrium measure, Newtonian potential 57
Equilibrium potential, 42
Equilibrium potential, Green 190
Equilibrium potential, linear 84
Equilibrium potential, logarithmic 76
Equilibrium potential, Newtonian 57
Equilibrium problem, Green potential 190
Equilibrium problem, logarithmic potential 213
Event 2
Exit distribution 88
Exit time 40
Exit time, last 23 200
Function, boundary 87
Function, concave 129
Function, continuous 16
Function, continuous, essentially 89
Function, continuous, extended sense 16
Function, continuous, Hoelder 115
Function, excessive 135
Function, Green 111
Function, harmonic 85
Function, locally integrable 85
Function, reduced 176
Function, resolutive 148
Function, semicontinuous, lower 19
Function, semicontinuous, upper 19
Function, subharmonic 128
Function, superharmonic 128
Function, superinvariant 135
Fundamental identity, -potential 41
Fundamental identity, Green potential 161
Fundamental identity, linear potential 83
Fundamental identity, logarithmic potential 71
Fundamental identity, Newtonian potential 55
Generalized 114
Harmonic continuation 107
Harmonic minorant 140
Harmonic minorant, greatest 140
Harnack's inequality 104 105
Harnack's theorem 105
Hitting distribution 28 160
Hitting distribution, starting at infinity 77
Hitting time 10
Hyperplane 29
Invariance, orthogonal 5
Invariance, scale 5
Invariance, translation 5
inversion 100
Karamata's Tauberian theorem 126
Kelvin transformation 101
Kolmogorov Extension Theorem 3
Laplacian 86
Laplacian, generalized 114
Limit at infinity 93
Lower class 146 148
Lower regularization 129
Markov property 7
Markov property, strong 12
Maximum principle, -potential 42
Maximum principle, Green potential 163
Maximum principle, logarithmic potential 207
Measure, Radon 17
Measure, Radon, signed 181
Measure, restriction 160
Measure, set of concentration 17
Measure, support 17
Monotone class theorem 5
Normal density 1
Point, cutoff 47
Point, irregular 30
Point, of thinness 141
Point, regular 30
Point, regular, for Dirichlet problem 89
Poisson equation 114
Poisson integral formula 104
Poisson integral representation 108
Potential 19
Potential, 41
Potential, Green 157
Potential, linear 82
Potential, logarithmic 71
Potential, Newtonian 54
Process, recurrent 25
Process, transient 25
PWB method 145
Quasi-e very where 174 204
Random variable 2
Reduite 176
Riesz decomposition theorem, Green potential 168
Riesz decomposition theorem, logarithmic potential 208
Robin constant, linear potential 84
Robin constant, logarithmic potential 76
Semigroup property 1 18
Set, 11
Set, -null 88
Set, capacitable, Green potential 203
Set, capacitable, logarithmic potential 226
Set, complete 186
Set, complete, weakly 186
Set, dense 184
Set, equilibrium 190
Set, Greenian 111
Set, nonpolar 20
Set, polar 20 144
Set, recurrent 24
Set, regular 89
Set, relatively compact 90
Set, starshaped 27
Set, starshaped, strongly 98
Set, thin 141
Set, transient 24
Set, transient, relatively 200
Shift transformation 7
Stopping time 8
Terminal time property 11
Thinness 141
Thorn 68
Transition operator, Brownian motion on open set 40
Transition operator, killed Brownian motion 33
Uniform probability distribution 16
Uniqueness principle, Green potential 175
Uniqueness principle, logarithmic potential 73 211
Uniqueness principle, Newtonian potential 55
Upper class 146 148
Wiener's test 66 96 145 225
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