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| Авторизация |
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| Dynkin E.B., Yushkevich A.A. — Markov processes; theorems and problems |
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| Предметный указатель |
 — Neighborhoods 50
2-Row phase space 127
Absorbing boundary 180 200
Absorption 75 160 147
Absorption, coefficient 190
Accessible boundary 173 197
Accession of points in optimal choice 90
Analysis and Markov processes, v 38
Attracting boundary 161 173
Birth and death process 151 154 164
Birth and death process, continued 174 189 200
boundary conditions 147 148 176 198
Boundary points (regular and irregular) 51 59 148 176 216
Boundary, absorbing 180 200
Boundary, accessible 173 197
Boundary, attracting 161 173
Boundary, classification of types 173 211
Boundary, finite 211
Boundary, inaccessible 173
Boundary, inward passable 196
Boundary, irregular 59
Boundary, nonabsorbing 217
Boundary, regular 51 54
Boundary, repelling 161 173
Boundary, strongly finite 211
Boundary, weakly finite 211
Brownian motion 41 52 65
Canonical coordinate 157
Cantor perfect set 196
Capacity 17 36
Capacity, definition 18
Carrier of set 35
Cauchy distribution 61
Cayley lottery problem 131
Characteristic operator 72 147
Characteristic operator, “class A” processes 176 189 209
Characteristic, birth and death processes 167
Characteristic, infinitesimal 69
Charge 18
Charge, maximum total 19
Chebyshev inequality 181
Choquet theorem 141
Circular domain 44 64
Compacmm 37 138
Compact convex sets 32 37
Concave function 113 122 225
Concave function, fundamental property 119
Conditional expectation of first-exit time 178
Conditional probability 88 92 96 213
Continuations of birth and death process 174 189 200
Continuous analog of symmetric random walk 113
Continuous random walk 39 113 233
Cumulative jump (transition) time 154 170
Decision 87 90
Denumerable Markov chain 119 136
Denumerable phase space 109 136
Derivative on a measure 170
Difference function 121
Diffusion processes 74 147 155 178
Dirichlet problem 33 48 60 114
Dirichlet problem, probabilistic treatment vi
Discrete potential 11 28 103
Distribution at time of exit from circle 43
Distribution,  -distribution of Wiener process 42
Distribution, Cauchy 67
Distribution, equilibrium 18
Distribution, limiting 40
Distribution, normal 40
Distribution, normed-sum 39
Distribution, Poisson 218
Domination principle 35
Double-row phase space 127
Elliptic operator L (diffusion process) 74
Envelope principle 35
Equilibrium distribution 18
Equilibrium potential 18
Escape probability 66 155 158
Escape probability from arbitrary domain 47
Escape probability from circle 43
Escape probability, harmonicity of 47
Escape probability, one-dimensional 75
Events (past, current, and future) 56
Excessive functions 15 36 102 138
Excessive functions, definition for multidimensional Wiener process 126
Excessive functions, denumerable Markov chain 136
Excessive majorant of payoff function 111 134
Exit time from arbitrary domain 84
Exit time from circle 43
Exit time from interval 164 210
Expectation (mathematical) 5 100 163 176
Expectation (mathematical), asymptotic expression 7 221
Expectation (mathematical), conditional, of first-exit time 178
Expectation (mathematical), incompletely defined random variables 41
Expectation (mathematical), limit of argument 169
Expectation (mathematical), number of hits at a point 5
Expectation (mathematical), sojourn time 195
Extinction 147 181
Extinction, coefficient 199
Extinction, probability 92
Extremal compact subset of set 31
Extremal points of set 31 37 141 143
Fee fa a game 134
Finite boundary 211
Finite Markov chain 92 100
Finite phase space 92 100
Fluid flow in a plane 70
Free group 142
Fubini’s Theorem 204
Game theory 100 134
General Markov chain 133
Generator of diffusion process (elliptic operator) 74
Green's formula 222
Green's function 136 222
Group (free) 142
Harmonic functions 7 37 60
Harmonic functions, bounded 8
Harmonic functions, definition 7
Harmonic functions, extended definition 136
Harmonic functions, nonnegative 15 61
Harmonic functions, positive 32 144
Helly theorem 139
Homogeneous Markov chain 91
HQ lder continuity 74
Inaccessible boundary 173
Induction (retrograde) 95 131 132
Infinite Markov chain 108
Infinite phase space 108
Infinitesimal operator 70
Inward passability (impassability) of boundary 196
Irregular boundary points 51 59
Jacobian 49 224
Jump measure 182 184
Jump probability 92 212
Laplace equation 18 62 114
Laplace operator 8 73 147 177 222
Laplace operator and Wiener processes 69 171
Lattice 2 19 30
Lattice parameter 39
Lattice “one-dimensional” 100
Left reflection 75
Linear functions 122
Local time 216
Lottery problem 131
Majorizadon of payoff function 105
Mapping of continuous functions into continuous functions (operator) 205
Markov chain 91 156
Markov chain, denumerable 119 136
Markov chain, general 133
Markov chain, homogeneous 91
Markov chain, infinite states 108
| Markov chain, recurrent 101
Markov processes 201 (see also Birth and death process “Class Diffusion Markov Random Wiener
Markov processes, relationship to Dirichlet problem vi
Markov property 46 70 120 148 152 202
Markov times 47 100 103 134 205 217
Martin boundary 136 143
Martin kernel 137
Maximal points (in selection problem,) 89
Mean exit time from arbitrary domain 84
Mean exit time from circle 43 84
Measurability considerations 42 49 114 126
Measure 41 50 137 139 170 182
Multidimensional random walk 2 40
Multidimensional Wiener process 118 1 26
Multiple-row phase space 129
Newtonian potential 11 17
Nonabsorbing boundary 217
Nondifferentiability of Wiener path 81
Nonrecurrent (transient) random walk 2
Nonrecurrent set 16
Normal distribution 40
Normed-sum distribution 39
One-dimensional random walk 1
One-parameter subgroups 121
One-step shift operator 7 103 121
Operators 69 73 121
Operators, characteristic 72 147
Operators, elliptic 74
Operators, infinitesimal 70
Operators, Laplace 8 69 147 171 177 222
Operators, mapping of continuous functions 205
Operators, one-step shift 7 103 121
Operators, resolvent, of process 203
Operators, velocity field 70
Operators, weak infinitesimal 71
Optimal choice problem 87 110
Optimal choice problem further generalization(s) of n “best” objects 129
Optimal stopping problem 87
Optimal stopping problem, general Maikov chain 133
Optimal stopping problem, Markov chain 98
Optimal stopping problem, sequence of independent (dependent) random variables 131
Optimal stopping problem, Wiener process 112
Optimal strategy 107
Path of particle in Brownian motion 41 52 65
Path of particle in random walk 5 22
Path of particle in Wiener process 41 81
Payoff function 14
Payoff function, bounded 101 105 127
Payoff function, unbounded 135
Phase space 155
Phase space, denumerable 109 136
Phase space, double-row 127
Phase space, finite 92 100
Phase space, infinite 108
Phase space, s-row 129
Plausible hypothesis 107
Poisson distribution 218
Poisson equation 171
Poisson equation, probabilistic solution 68
Poisson flow 218 219
Poisson integral 68 84
Potential 11
Potential,  -potentials 123
Potential, discrete 11 28 103
Potential, equilibrium 18
Potential, extended definition 136
Potential, Newtonian 11 17
Potential, probabilistic interpretation 12
Probabilistic solution of equations 39
Probability in analysis 39 147
Probability of continued birth and death process 175
Probability of given state at first-exit time from interval 158
Probability of independent events 58
Probability of no return to starting point 63
Probability of random walk event 2
Probability of rated objects 88
Probability of success (payoff) 88 93 102 129
Probability of visit to set after nth step 7 16
Probability of visit to set infinitely often after first step 10
Probability total 1 12 94
Probability transition 3 42 46 92
Probability, conditional 88 92 96 213
Probability, escape (see Escape probability)
Probability, extinction 92
Probability, jump 92 212
Random walk on a free group with finite number of generators 142
Random walk on a lattice 2 30 39 60 98 100 103 151
Random walk on phase space 92
Random walk one-dimensional 1
Random walk with absorption 110
Random walk, continuous 39
Random walk, continuous analog of symmetric 113
Random walk, multidimensional 2 40
Random walk, nonrecurrent (transient) 2
Random walk, recurrent 2
Random walk, symmetric 1 103 113
Random walk, unsymmetric 37
Rebound at boundary 148
Recurrence of coordinate plane 25
Recurrence of Markov chain 101 162
Recurrence of sets 1 64
Recurrence on axis 25
Recurrence, alternate definition 10 21
Recurrence, criterion 19
Recurrence, definition 2
Recurrent set 16 (see also Recurrence)
Reflecting barrier 211
Reflection 75 147 214
Reflection time 214
Reflection, coefficient 182 184 190
Reflection, “x-reflection” 214 220
Regular boundary points 51 59
Regularity 51 56 59
Regularity, necessary and sufficient conditions 82
Regularity, strong sufficient condition 83
Repelling boundary 161 173
Resolvent (operator) 203
Retrograde induction 95 131 132
Right reflection 76
s-Row phase space 129
Selection problem (see Optimal choice problem)
Selection strategy (optimal) 88 100
Selection strategy (optimal), table 97
Self-similariiy of Wiener process 44
Sets, -support 109 119 134 135
Sets, Cantor perfect 196
Sets, compact convex 32 37
Sets, convex 31 141
Sets, nonrecurrent 16
Sets, recurrent 16
Sets, support 107 129 131
Sets, unpredictable 69
Sets, “sparse” on lattice 64
Sojourn time in state 165 184 194
stopping 87 95 112 131
Stopping, instantaneous 106
Strong Markov properly 46 124 177 203 212
Success (payoff) probability 5 88 93 102 129
Successive approximations, method of 134
Superharmonic functions 8 119 126
Superharmonic functions, definition 8
Superharmonic functions, nonnegative 15
Support set 107 129 131
Support set, -support set 109 119 134 135
Sweeping-out principle 35
Symmetric random walk 1 103 113
Time of first exit from circle 61
Time of first exit from zero 54
Time of first visit to circle 81
Time of first visit to layer 24
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