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Perturbation of functions by the paths of a Lévy process

Published online by Cambridge University Press:  24 October 2008

Steven N. Evans
Steven N. Evans
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.
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Extract

In a recent paper Mountford [4] showed, using an ingenious probabilistic argument, that if X is a real-valued stable process with index α < 1 and f: [0, ∞) → ℝ is a non-constant continuous function, then

where we use the notation |A| for the Lebesgue measure of a Lebesgue measurable set A ⊂ ℝ. The argument in [4] appears to make strong use of the strict scaling properties of X and the ‘intermediate value’ property of f.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 2 , March 1989 , pp. 377 - 380
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Blumenthal, R. M. and Getoor, R. K.. Markov Processes and Potential Theory (Academic Press, 1968).Google Scholar
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[3]Fristedt, B.. Sample functions of stochastic processes with stationary independent increments. Adv. Probab. Related Topics. 3 (1974), 241–396.Google Scholar
[4]Mountford, T. S.. Time inhomogeneous Markov processes and the polarity of single points. Ann. Probab. (To appear.)Google Scholar
[5]Serrin, J. and Varberg, D. E.. A general chain rule for derivatives and the change of variable formula for the Lebesgue integral. Amer. Math. Monthly 76 (1969), 514–520.CrossRefGoogle Scholar