Abstract
The purpose of this paper is to present in a more or less self-contained way the chief facts about the local times \(\mathfrak{t}\) of one-dimensional Brownian motion due to P. Lévy, F. Knight, D. B. Ray, and Itô-McKean. The deepest part concerns the remarkable fact that for a class of stopping times \(\mathfrak{m}\), such as passage times and independent exponential holding times, the local time \(\mathfrak{t}(\mathfrak{m},x)\) is a diffusion relative to its spatial parameter x. The beautiful methods of D. Williams are employed here as being most in the manner of P. Lévy who began the whole thing. The intent is purely expository, and only the main features of the proofs are indicated. A familiarity with the most elementary facts about Brownian motion is assumed. The paper is dedicated to Norman Levinson with affection and respect.
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Notes
- 1.
D. Williams [May 8, 1972] writes: “I still cannot find a sane proof of [this.]—To get a sane proof, I formulated a Splitting Time Theorem that good Markov processes start afresh at ‘splitting times’ (but with new laws). You of all people should recognize the terminology! M. Jacobsen (visiting here from Copenhagen) has a nice Galmarino-type definition of splitting times, but we cannot get near the theorem. All the \(\mathfrak{l}\)’s and \(\mathfrak{m}\)’s in your paper are splitting times so the result would be useful.”
References
L. Breiman. Probability. Addison-Wesley, Massachussetts, 1968.
K. Itô and H. P. McKean. Brownian motion on a half line. Ill. J. Math., 7:181–231, 1963.
K. Itô and H. P. McKean. Diffusion Processes and Their Sample Paths. Academic Press, 1965.
F. B. Knight. Random walks and a sojourn density process for Brownian motion. Trans. AMS, 167:56–86, 1963.
P. Lévy. Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris, 1948.
H. P. McKean. Excursions of a non-singular diffusion. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1:230–239, 1963.
H. P. McKean. Stochastic Integrals, volume 5 of Probability and Mathematical Statistics. Academic Press, 1969.
D. B. Ray. Sojourn times of a diffusion process. Illinois J. Math., 7:615–630, 1963.
M. L. Silverstein. A new approach to local times. J. Math. Mech., 17:1023–1054, 1968.
A. V. Skorohod. Stochastic equations for diffusion processes with a boundary. Teor. Verojatnost. i Primenen, 6:287–298, 1961.
H. F. Trotter. A property of Brownian motion paths. Illinois J. Math., 2:425–433, 1958.
D. Williams. Markov properties of Brownian local time. Bull. Amer. Math. Soc., 75:1035–1036, 1964.
D. Williams. Decomposition of the Brownian path. Bull. Amer. Math. Soc., 70:871–973, 1970.
D. Williams. Markov properties of Brownian local time (2). Unpublished, 1971.
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Dedicated To Norman Levinson
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McKean, H.P. (2015). Brownian Local Times. In: Grünbaum, F., van Moerbeke, P., Moll, V. (eds) Henry P. McKean Jr. Selecta. Contemporary Mathematicians. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22237-0_14
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