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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

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