## Elements of the Theory of Markov Processes and Their ApplicationsGraduate-level text and reference in probability, with numerous scientific applications. Nonmeasure-theoretic introduction to theory of Markov processes and to mathematical models based on the theory. Appendixes. Bibliographies. 1960 edition. |

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Page 113

It can be shown that F.(81,82,t) satisfies the differential equation o = f(F1, Fe) i = 1

, 2 (2.230) with

equation 9F 9F 9F + = f(81.8,) + +f,(s1,82) + (2.232) 9t 08.1 0s, From (2.230) and

...

It can be shown that F.(81,82,t) satisfies the differential equation o = f(F1, Fe) i = 1

, 2 (2.230) with

**initial condition**F.(81.82,0) = 8, (2.231) or the partial differentialequation 9F 9F 9F + = f(81.8,) + +f,(s1,82) + (2.232) 9t 08.1 0s, From (2.230) and

...

Page 138

where A, B, and A, are determined by the

must be specified for any particular diffusion problem. We now consider the

application of the Laplace transformation" to the Kolmogorov diffusion equations.

where A, B, and A, are determined by the

**initial**and boundary**conditions**whichmust be specified for any particular diffusion problem. We now consider the

application of the Laplace transformation" to the Kolmogorov diffusion equations.

Page 389

which is to be solved with the

Eq. (9.34) that satisfies the above

t)} X | - 'so {—st + [I — p(s)]A(t)}F(t,0) a. (9.35) 0 where F(t,0) is the probability that

...

which is to be solved with the

**initial condition**p(0,8) = 1. The unique solution ofEq. (9.34) that satisfies the above

**initial condition**is p(t,3) = exp (st — [1 — p(s)]A(t)} X | - 'so {—st + [I — p(s)]A(t)}F(t,0) a. (9.35) 0 where F(t,0) is the probability that

...

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### Contents

Introduction | 1 |

Processes Discrete in Space and Continuous in Time | 57 |

Processes Continuous in Space and Time | 129 |

Copyright | |

9 other sections not shown

### Other editions - View all

Elements of the Theory of Markov Processes and Their Applications A. T. Bharucha-Reid Limited preview - 2012 |

Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |

### Common terms and phrases

absorber addition applications approach arrival associated assume assumptions becomes birth boundary branching processes called cascade coefficients collision concerned condition consider constant continuous counter death defined denote density derive described determined developed differential equation diffusion discussion distribution function electron energy epidemic equal exists expected expression finite fluctuation given gives growth Hence independent individuals initial condition integral interest interval introduce Kolmogorov equations Laplace transform length limit machine Markov Markov chain Markov processes Math mathematical mean method moments necessary nucleon obtain particle particular photon Poisson population positive primary problem Proof properties queueing radiation random variable reaction refer relation represent respectively satisfies shown simple ſº solution Statist Stochastic Processes Theorem theory tion transition probabilities zero