On Wiener process sample paths

Abstract

Let { X t ( ω ) } \{ {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 ( ω ) } \{ {X_t}(\omega )\} can be modified so that almost surely the sample paths are everywhere locally recurrent.