## Continuous Semi-Markov ProcessesThis title considers the special of random processes known as semi-Markov processes. These possess the Markov property with respect to any intrinsic Markov time such as the first exit time from an open set or a finite iteration of these times. The class of semi-Markov processes includes strong Markov processes, Lévy and Smith stepped semi-Markov processes, and some other subclasses. Extensive coverage is devoted to non-Markovian semi-Markov processes with continuous trajectories and, in particular, to semi-Markov diffusion processes. Readers looking to enrich their knowledge on Markov processes will find this book a valuable resource. |

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### Contents

Introduction | 9 |

Stepped SemiMarkov Processes | 17 |

Sequences of First Exit Times and Regeneration Times | 37 |

Copyright | |

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additive functional admissible family asymptotics coefficients compact set completely monotone condition consider construction continuous semi-Markov process convergence coordinates corresponding curvilinear integral deducing sequence denote determined differential equation diffusion type Dirichlet problem distribution Dynkin's formulae equal equation 5.6 exists exit family of additive family of measures family of probability finite formula Hence infinitesimal operator initial point intervals of constancy kernel lambda-characteristic operator lambda-continuous Laplace family Laplace transformation lemma Let Px limit Markov chain Markov process Markov property measurable function measures Px metric metric space monotone function neighborhood obtain open sets parameter point of discontinuity probability measures process with independent Proof proposition proved pseudo-local regeneration representation semi-Markov property sigma-algebra SM process solution space stepped semi-Markov process subsets tends to zero theorem trace trajectory transition functions transition generating functions weak convergence