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AuthorBlumenthal, Robert M. author
TitleExcursions of Markov Processes [electronic resource] / by Robert M. Blumenthal
ImprintBoston, MA : Birkhรคuser Boston, 1992
Connect tohttp://dx.doi.org/10.1007/978-1-4684-9412-9
Descript XII, 276 p. online resource

SUMMARY

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 applyยญ ing esoteric things like David Williams' path decomposition, it provides a method for constructing processes whose description in terms of an inยญ 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


CONTENT

I 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 Itรดโ{128}{153}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. Rayโ{128}{153}s local time theorem -- 3. Trotterโ{128}{153}s theorem -- 4. Super Brownian motion -- VII Excursions Away From a Set -- 1. Introduction -- 2. Additive functionals and Lรฉvy systems -- 3. Exit systems -- 4. Motoo Theory -- Notation Index


Mathematics Probabilities Mathematics Probability Theory and Stochastic Processes



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