03673nam a22003375i 4500978-1-4684-9412-9cr nn 008mamaa121227s1992 xxu: s :::: 0:eng d9781468494129ENGUSQA274-274.9519.2Online resource : BirkhäuserBlumenthal, Robert McCallum1931-Excursions of Markov Processes[EBook]by Robert M. Blumenthal.Boston, MABirkhäuser1992.XII, 276 pagesonline resource.textonline resourceProbability and Its ApplicationsI Markov Processes -- 0. Introduction -- 1. Basic terminology -- 2. Stationary transition functions -- 3. Time homogeneous Markov processes -- 4. The strong Markov property -- 5. Hitting times -- 6. Standard processes -- 7. Killed and stopped processes -- 8. Canonical realizations -- 9. Potential operators and resolvents -- II Examples -- 1. Examples -- 2. Brownian motion -- 3. Feller Brownian motions and related examples -- III Point Processes of Excursions -- 1. Additive processes -- 2. Poisson point processes -- 3. Poisson point processes of excursions -- IV Brownian Excursion -- 1. Brownian excursion -- 2. Path decomposition -- 3. The non-recurrent case -- 4. Feller Brownian motions -- 5. Reflecting Brownian motion -- V Itô’s Synthesis Theorem -- 1. Introduction -- 2. Construction -- 3. Examples and complements -- 4. Existence and uniqueness -- 5. A counter-example -- 6. Integral representation -- VI Excursions and Local Time -- 1. Introduction -- 2. Ray’s local time theorem -- 3. Trotter’s theorem -- 4. Super Brownian motion -- VII Excursions Away From a Set -- 1. Introduction -- 2. Additive functionals and Lévy systems -- 3. Exit systems -- 4. Motoo Theory -- Notation Index.Let {Xti t ~ O} be a Markov process in Rl, and break up the path X t into (random) component pieces consisting of the zero set ({ tlX = O}) and t the "excursions away from 0," that is pieces of path X. : T ::5 s ::5 t, with Xr- = X = 0, but X. 1= 0 for T < s < t. When one measures the time in t the zero set appropriately (in terms of the local time) the excursions acquire a measure theoretic structure practically identical to that of processes with stationary independent increments, except the values of the process are paths rather than real numbers. And there is a measure on path space that helps describe the measure theoretic properties of the excursions in the same way that the Levy measure describes the jumps of a process with independent increments. The entire circle of ideas is called excursion theory. There are many attractive things about the subject: it is an area where one can use to advantage general probabilistic potential theory to make quite specific calculations, it provides a natural setting for apply ing esoteric things like David Williams' path decomposition, it provides a method for constructing processes whose description in terms of an in finitesimal generator or some such analytic object would be complicated. And the ideas seem to be closely related to a good deal of current research in probability.Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)Markov processesProbabilities.Mathematics.Probability Theory and Stochastic Processes.SpringerLink (Online service)Probability and Its Applicationshttp://dx.doi.org/10.1007/978-1-4684-9412-9