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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the continuity of local times


Author: Donald Geman
Journal: Proc. Amer. Math. Soc. 57 (1976), 321-326
MSC: Primary 60G17; Secondary 60G15
DOI: https://doi.org/10.1090/S0002-9939-1976-0420812-4
MathSciNet review: 0420812
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Abstract: Several conditions are given for a stochastic process $ X(t)$ on $ [0,1]$ to have a local time which is continuous in its time parameter (for example, in the Gaussian case, the integrability of $ {[E{(X(t) - X(s))^2}]^{ - 1/2}}$ over the unit square). Furthermore, for any Borel function $ F$ on $ [0,1]$ with a continuous local time, the approximate limit of $ \vert F(s) - F(t)\vert/\vert s - t\vert$ as $ s \to t$ is infinite for a.e. $ t \in [0,1]$ and $ s\vert F(s) = F(t) $ is uncountable for a.e. $ t \in [0,1]$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0420812-4
Article copyright: © Copyright 1976 American Mathematical Society

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