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Asymptotic Estimates Of The Green Functions And Transition Probabilities For Markov Additive Processes

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1.	Title	Title of document	Asymptotic Estimates Of The Green Functions And Transition Probabilities For Markov Additive Processes
2.	Creator	Author's name, affiliation, country	Kouhei Uchiyama; Tokyo Institute of Technology
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	asymptotic expansion, harmonic analysis, semi-Markov process, random walk with internal states, perturbation, aperiodicity, ergodic, Doeblin's condition
3.	Subject	Subject classification	60K15;60J45;60K35;60J05
4.	Description	Abstract	In this paper we shall derive asymptotic expansions of the Green function and the transition probabilities of Markov additive (MA) processes $(\xi_n, S_n)$ whose first component satisfies Doeblin's condition and the second one takes valued in $Z^d$. The derivation is based on a certain perturbation argument that has been used in previous works in the same context. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important for instance in computation of the harmonic measures of a half space for certain models. We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be clarified and it will be shown that in a simple manner the process, if not degenerate, are transformed to another one that satisfies Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if the MA processes is irreducible as a Markov process, then the Green function is expanded quite similarly to that of a classical random walk on $Z^d$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Monbukagakusho grand-in-aid no. 15540109
7.	Date	(YYYY-MM-DD)	2007-02-16
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/396
10.	Identifier	Digital Object Identifier	10.1214/EJP.v12-396
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 12
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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