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ОбложкаSchinazi R.B. Classical and spatial stochastic processes: with applications to biology. - 2nd ed. - New York: Springer Science + Business Media: Birkhauser, 2014. - xii, 268 p.: ill. - Incl. bibl. ref. - Ind.: p.267-268. - ISBN 978-1-4939-1868-3
Шифр: (И/В17-S32) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
1  A Short Probability Review ................................... 1
   1  Sample Space and Random Variables ......................... 1
      1.1  Sample Space and Probability Axioms .................. 1
      1.2  Discrete Random Variables ............................ 2
   2  Independence .............................................. 4
      2.1  Binomial Random Variables ............................ 5
      2.2  Geometric Random Variables ........................... 6
      2.3  A Sum of Poisson Random Variables .................... 7
   3  Conditioning .............................................. 8
      3.1  Thinning a Poisson Distribution ...................... 9
   4  Generating Functions ..................................... 10
      4.1  Sum of Poisson Random Variables (Again) ............. 12
      4.2  Thinning a Poisson Distribution (Again) ............. 13
      Problems ................................................. 13
      References ............................................... 15

2  Discrete Time Branching Process ............................. 17
   1  The Model ................................................ 17
      Problems ................................................. 21
   2  The Probability of a Mutation in a Branching Process ..... 24
      2.1  An Equation for the Total Progeny Distribution ...... 24
      2.2  The Total Progeny Distribution in a Particular
           Case ................................................ 25
      2.3  The Probability of a Mutation ....................... 28
      2.4  Application: The Probability of Drug Resistance ..... 29
      2.5  Application: Cancer Risk ............................ 32
      2.6  The Total Progeny May Be Infinite ................... 34
      Problems ................................................. 36
   3  Proof of Theorem 1.1 ..................................... 39
      Problems ................................................. 44
      Notes .................................................... 45
      References ............................................... 46

3  The Simple Symmetric Random Walk ............................ 47
   1  Graphical Representation ................................. 48
   2  Returns to 0 ............................................. 52
   3  Recurrence ............................................... 56
   4  Last Return to the Origin ................................ 60
      Problems ................................................. 62
      Notes .................................................... 65
      References ............................................... 65

4  Asymmetric and Higher Dimension Random Walks ................ 67
   1  Transience of Asymmetric Random Walks .................... 67
   2  Random Walks in Higher Dimensions ........................ 68
      2.1  The Two Dimensional Walk ............................ 68
      2.2  The Three Dimensional Walk .......................... 69
   3  The Ruin Problem ......................................... 71
      3.1  The Greedy Gambler .................................. 74
      3.2  Random Walk Interpretation .......................... 75
      3.3  Duration of the Game ................................ 75
      Problems ................................................. 77

5  Discrete Time Markov Chains ................................. 81
   1  Classification of States ................................. 81
      1.1  Decomposition of the Chain .......................... 82
      1.2  A Recurrence Criterion .............................. 84
      1.3  Finite Markov Chains ................................ 88
      Problems ................................................. 90
   2  Birth and Death Chains ................................... 91
      2.1  The Coupling Technique .............................. 96
      2.2  An Application: Quorum Sensing ...................... 98
      Problems ................................................ 100
      Notes ................................................... 103
      References .............................................. 103

6  Stationary Distributions for Discrete Time Markov Chains ... 105
   1  Convergence to a Stationary Distribution ................ 105
      1.1  Convergence and Positive Recurrence ................ 105
      1.2  Stationary Distributions ........................... 109
      1.3  The Finite Case .................................... 113
      Problems ................................................ 115
   2  Examples and Applications ............................... 117
      2.1  Reversibility ...................................... 117
      2.2  Birth and Death Chains ............................. 118
      2.3  The Simple Random Walk on the Half Line ............ 120
      2.4  The Ehrenfest Chain ................................ 122
      2.5  The First Appearance of a Pattern .................. 125
      Problems ................................................ 126
      Notes ................................................... 128
      References .............................................. 129

7  The Poisson Process ........................................ 131
   1  The Exponential Distribution ............................ 131
      Problems ................................................ 134
   2  The Poisson Process ..................................... 135
      2.1   Application: Influenza Pandemics .................. 145
      Problems ................................................ 146
      Notes ................................................... 148
      References .............................................. 149

8  Continuous Time Branching Processes ........................ 151
   1  A Continuous Time Binary Branching Process .............. 151
      Problems ................................................ 153
   2  A Model for Immune Response ............................. 155
      2.1  The Tree of Types .................................. 156
      Problems ................................................ 160
   3  A Model for Virus Survival .............................. 162
      Problems ................................................ 168
   4  A Model for Bacterial Persistence ....................... 169
      Problems ................................................ 172
      Notes ................................................... 173
      References .............................................. 173

9  Continuous Time Birth and Death Chains ..................... 175
   1  The Kolmogorov Differential Equations ................... 175
      1.1  The Pure Birth Processes ........................... 178
      1.2  The Yule Process ................................... 180
      1.3  The Yule Process with Mutations .................... 181
      1.4  Passage Times ...................................... 182
      Problems ................................................ 184
   2  Limiting Probabilities .................................. 186
      Problems ................................................ 193
      Notes ................................................... 195
      References .............................................. 195

10 Percolation ................................................ 197
   1  Percolation on the Lattice .............................. 197
      Problems ................................................ 202
   2  Further Properties of Percolation ....................... 205
      2.1  Continuity of the Percolation Probability .......... 205
      2.2  The Subcritical Phase .............................. 208
      Problems ................................................ 212
   3  Percolation On a Tree and Two Critical Exponents ........ 214
      Problems ................................................ 216
      Notes ................................................... 217
      References .............................................. 218

11 A Cellular Automaton ....................................... 219
   1  The Model ............................................... 219
      Problems ................................................ 222
   2  A Renormalization Argument .............................. 224
      Problems ................................................ 228
      Notes ................................................... 229
      References .............................................. 229

12 A Branching Random Walk .................................... 231
   1  The Model ............................................... 231
      1.1  The Branching Random Walk on the Line .............. 233
      1.2  The Branching Random Walk on a Tree ................ 235
      1.3  Proof of Theorem 1.1 ............................... 237
      Problems ................................................ 242
   2  Continuity of the Phase Transitions ..................... 243
      2.1  The First Phase Transition Is Continuous ........... 243
      2.2  The Second Phase Transition Is Discontinuous ....... 246
      Problems ................................................ 249
      Notes ................................................... 249
      References .............................................. 249

13 The Contact Process on a Homogeneous Tree .................. 251
   1  The Two Phase Transitions ............................... 251
      Problems ................................................ 253
   2  Characterization of the First Phase Transition .......... 254
      Problems ................................................ 255
      Notes ................................................... 256
      References .............................................. 256

A  A Little More Probability .................................. 257
   1  Probability Space ....................................... 257
      Problems ................................................ 260
   2  Borel-Cantelli Lemma .................................... 261
      2.1  Infinite Products .................................. 263
      Problems ................................................ 264
      Notes ................................................... 265
      References .............................................. 265

Index ......................................................... 267

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