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Characterization of the Levy Measures of Inverse Local Times of Gap Diffusion

  • Frank B. Knight
Chapter
Part of the Progress in Probability and Statistics book series (PRPR, volume 1)

Abstract

Let X(t) be a persistent nonsingular diffusion on an interval Q containing 0 (in the sense of [4]). If we assume a natural scale, then X is characterized by its speed measure m(dx) on Q, and finite endpoints are reflecting. There exists (P0-a.s.) the local time
$$ l(t) = \mathop{{\lim }}\limits_{{{h_i} \to 0 + }} \frac{1}{{m[ - {h_1},{h_2})}}\int\limits_0^t {{I_{{[ - {h_1},{h_2})}}}(X(s))ds} $$
(1.1)
which is continuous in t ≥ 0. The right-continuous inverse ℓ(-1)(t) = inf{s: ℓ(s) > t} is an increasing finite process with homogeneous independent increments, and as such (see for example [4, §6.1]) it is characterized by its Lévy measure n(dy) on (0,∞) through the equation
$$ E\exp [ - \lambda {l^{{( - 1)}}}(t)] = \exp [ - t{m_0}\lambda - t\int\limits_0^{\infty } {(1 - {e^{{ - \lambda y}}})n(dy)],\lambda > 0} $$
where m0 = m{0}. An interesting problem [4, p. 217] is to characterize the class of all n(dy) which can appear when Q and m(dx) vary. It is shown there that there is a unique measure µ on [0,∞) such that \( n(dy) = dy\int_{{0 - }}^{\infty } {{e^{{ - y\gamma }}}\mu (d\gamma )} \), so the problem reduces to characterizing the class of µ.

Keywords

Local Time Speed Measure Natural Scale Negative Excursion Hunt Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H. DYM and H.P. McKEAN. Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic Press, New York, 1976.Google Scholar
  2. 2.
    W. Feller. Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1 (1957), 459–504.Google Scholar
  3. 3.
    H. GZYL. Lévy systems for time-changed processes. The Annals of Probability 5 (1977), 565–570.CrossRefGoogle Scholar
  4. 4.
    K. ITO and H.P. McKEAN Jr. Diffusion processes and their sample paths. Academic Press, New York, 1965.Google Scholar
  5. 5.
    I.S. KAC and M.G. KREIN. On the spectral function of the string. Amer. Math. Society Translations 2, Vol. 103 (1974), 19–102.Google Scholar
  6. 6.
    J.F.C. KINGMAN. Homecomings of Markov processes. Adv. Appl. Probability 4 (1973), 66–102.CrossRefGoogle Scholar
  7. 7.
    F.B. KNIGHT and A.O. PITTENGER. Excision of a strong Markov process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 23 (1972), 114–120.CrossRefGoogle Scholar
  8. 8.
    F.B. KNIGHT. Essentials of Brownian Motion and Diffusion. Mathematical Surveys No. 18. American Mathematical Society, Providence, 1981.Google Scholar
  9. 9.
    M.G. KREIN and NUDEL’MAN. The Markov Moment Noblem and Extremal Problems. Translations of math. monographs, Vol. 50. American Mathematical Society, Providence, 1977.Google Scholar
  10. 10.
    U. KüCHLER. Some asymptotic properties of the transition densities of one-dimensional quasi-diffusions. Publ. R.I.M.S. Kyoto Univ. 16 (1980), 245–268.CrossRefGoogle Scholar
  11. 11.
    S. WATANABE. On time inversion of one-dimensional diffusion processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31 (1975), 115–124.CrossRefGoogle Scholar
  12. 12.
    D. WILLIAMS. Diffusions, Markov Processes, and Martingales, Vol. 1. J. Wiley and Sons, London, 1979.Google Scholar

Copyright information

© Birkhäuser Boston 1981

Authors and Affiliations

  • Frank B. Knight
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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