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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On Wiener process sample paths

Authors: G. J. Foschini and R. K. Mueller
Journal: Trans. Amer. Math. Soc. 149 (1970), 89-93
MSC: Primary 60.62
MathSciNet review: 0258129
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Abstract: Let $ \{ {X_t}(\omega )\} $ represent a version of the Wiener process having almost surely continuous sample paths on $ ( - \infty ,\infty )$ that vanish at zero. We present a theorem concerning the local nature of the sample paths. Almost surely the local behavior at each t is of one of seven varieties thus inducing a partition of $ ( - \infty ,\infty )$ into seven disjoint Borel sets of the second class. The process $ \{ {X_t}(\omega )\} $ can be modified so that almost surely the sample paths are everywhere locally recurrent.

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Keywords: Wiener process, sample paths, local behavior, Borel sets, oscillation from the right, increase from the left, local maxima, everywhere locally recurrent functions
Article copyright: © Copyright 1970 American Mathematical Society