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Basic Properties

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Part of the Grundlehren der mathematischen Wissenschaften book series (GL,volume 249)

Abstract

Let a homogeneous Markov process {X t , 퓕 t , t ∈ T} with transition semigroup (P t ) be given. We seek a class of functions f on E such that {f(X t ), 퓕1,} is a supermartingale.

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Notes on Chapter 2

  1. This chapter serves as an interregnum between the more concrete Feller processes and Hunt’s axiomatic theory. It is advantageous to introduce some of the basic tools at an early stage.

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  2. §2.2. Feller process is named after William Feller who wrote a series of pioneering papers in the 1950’s. His approach is essentially analytic and now rarely cited. The sample function properties of his processes were proved by Kinney, Dynkin, Ray, Knight, among others. Dynkin [1] developed Feller’s theory by probabilistic methods. His book is rich in content but difficult to consult owing to excessive codification. Hunt [1] and Meyer [2] both discuss Feller processes before generalizations.

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  3. It may be difficult for the novice to appreciate the fact that twenty five years ago a formal proof of the strong Markov property was a major event. Who is now interested in an example in which it does not hold? A full discussion of augmentation is given in Blumenthal and Getoor [1]. This is dry and semi-trivial stuff but inevitable for a rigorous treatment of the fundamental concepts. Instead of beginning the book by these questions it seems advisable to postpone them until their relevance becomes more apparent.

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  4. There is some novelty in introducing the moderate Markov property before quasi left continuity; see Chung [5]. It serves as an illustration of the general methodology alluded to in §1.3, where both F T+ and FT- are considered. Historically, a moderate Markov property was first observed at the “first infinity” of a simple kind of Markov chains, see Chung [3]. It turns out that a strong Markov process becomes moderate when the paths are reversed in time, see Chung and Walsh [1]. A more complete discussion of the measurability of hitting times will be given in §3.3. Hunt practically began his great memoir [1] with this question, fully realizing that the use of hitting times is the principal method in Markov processes.

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© 1982 Springer Science+Business Media New York

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Chung, K.L. (1982). Basic Properties. In: Lectures from Markov Processes to Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol 249. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1776-1_2

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  • DOI: https://doi.org/10.1007/978-1-4757-1776-1_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4757-1778-5

  • Online ISBN: 978-1-4757-1776-1

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