Abstract
In his 1939 paper [1] Lévy introduced the notion of local time for Brownian motion. He gave several equivalent definitions, and towards the end of that long paper he proved the following result. Let ∈ > 0, t > 0, B(0) = 0,
where B(t) is the Brownian motion in R and m is the Lebesgue measure. Then almost surely the limit below exists for all t >0:
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References
P. LÉVY, Sur certains processus stochastiques homogènes. Compositio Math. 7 (1939), 283–339.
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K.L. CHUNG, Reminiscences of some of Paul Lévy’s Ideas in Brownian Motion and in Malloy Chains. Seminar on Stochastic Processes 1988, 99–108. Birkhäuser, 1989. Also printed with the author’s permission but without the Postscript in Colloque Paul Lévy, Soc. Math. de France, 1988.
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Balkema, A.A., Chung, K.L. (1991). Paul Lévy’s Way to His Local Time. In: Çinlar, E., Fitzsimmons, P.J., Williams, R.J. (eds) Seminar on Stochastic Processes, 1990. Progress in Probability, vol 24. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-0562-0_2
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