Hostname: page-component-54dcc4c588-54gsr Total loading time: 0 Render date: 2025-09-27T22:37:21.568Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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].

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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