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Some eigenvalue identities for Brownian motion

Published online by Cambridge University Press:  04 October 2011

Paul McGill
Paul McGill
Affiliation:
School of Theoretical Physics, DIAS, 10 Burlington Road, Dublin 4, Ireland
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Extract

The general problem can be stated as follows. Take a Brownian motion Bt started at − x < 0, and consider the additive functional At = ∫L(a,t)m(da), where L(a, t) is the Brownian local time. We suppose that m = m+ — m−, where these are positive measures supported respectively on (0, ∞) and (— ∞, 0). Then, with the equalization time defined by T = inf {t > 0: At = 0}, we ask for an explicit evaluation of the law π (x,dy) = P−x[BT∈dy]. In [8, 9] we showed how π (x,dy) can be obtained by solving an integral convolution equation of Wiener-Hopf type. The method used there exploits a technique of Ray [10].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 587 - 596
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Baker, N.. Some integral equalities in Wiener-Hopf theory. In Stochastic Analysis and Applications. Lecture Notes in Math. vol. 1095 (Springer-Verlag, 1984), pp. 169–186.CrossRefGoogle Scholar
[2] Bingham, N.. Fluctuation theory in continuous time. Adv. in Appl. Prob. 7 (1975), 705–766.CrossRefGoogle Scholar
[3] Bingham, N.. Maximum of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 (1973), 273–296.CrossRefGoogle Scholar
[4] Erdelyi, A.. The Bateman Manuscript Project (Higher Transcendental Functions, vol. 2) (McGraw-Hill, 1953).Google Scholar
[5] Erdelyi, A.. The Bateman Manuscript Project (Integral transforms, vol. 1) (McGraw-Hill, 1953).Google Scholar
[6] Ito, K. and McKean, H. P.. Diffusion Processes and their Sample Paths (Springer-Verlag, 1965).Google Scholar
[7] London, R. R., McKean, H. P., Rogers, L. C. G. and Williams, D.. A martingale approach to some Wiener-Hopf Problems I. In Séminaire de Probabilités XVI, Lecture Notes in Math, vol. 920 (Springer-Verlag, 1982), pp. 41–67.CrossRefGoogle Scholar
[8] McGill, P.. Wiener-Hopf factorisation of Brownian motion. (Preprint, 1986.)Google Scholar
[9] McGill, P.. Wiener-Hopf factorisation of absorbing Brownian motion. (Preprint, 1986.)Google Scholar
[10] Ray, D. B.. Stable processes with absorbing barrier. Trans. Amer. Math. Soc. 89 (1958), 16–24.CrossRefGoogle Scholar