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The harmonic functions of (At, Bt,)1

Published online by Cambridge University Press:  24 October 2008

L. C. G. Rogers
L. C. G. Rogers
Affiliation:
School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 4NS
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Extract

The non-negative harmonic functions of a transient Markov process yield a great deal of information about the ‘behaviour at infinity’ of the process, and can be used to h-transform the process to behave in a certain way at infinity. The traditional analytic way of studying the non-negative harmonic functions is to construct the Martin boundary of the process (see, for example, Meyer [4], Kunita and T. Watanabe[3], and Kemeny, Snell & Knapp[2], Williams [7] for the chain case). However, certain conditions on the process need to be satisfied, one of the most basic of which is that there exists a reference measure η such that Uλ (x, ·) ≪ η for all λ > 0, all x ∈ E, the state space of the Markov process. (Here, (Uλ)λ>0 is the resolvent of the process.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 2 , September 1993 , pp. 369 - 377
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Hobson, D. G.. Large deviations of the are sine law. Preprint, 1992.Google Scholar
[2]Kemeny, J. G., Snell, J. L. and Knapp, A. W.. Denumerable Markov Chains (Van Nostrand, 1966).Google Scholar
[3]Kunita, H. and Watanabe, T.. Markov processes and Martin boundaries I. Ill. J. Math. 9 (1965), 485–526.Google Scholar
[4]Meyer, P. A.. Processes de Markov: la frontière Martin. Lecture Notes in Mathematics 77 (Springer-Verlag, 1970).Google Scholar
[5]Rogers, L. C. G. and Williams, D.. Diffusions, Markov Processes and Martingales. Vol. 2 (Wiley, 1987).Google Scholar
[6]Williams, D.. Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Soc. 28 (1974), 738–768.CrossRefGoogle Scholar
[7]Williams, D.. Diffusions, Markov Processes and Martingales. Vol. 1 (Wiley, 1979).Google Scholar