On Wiener Process Sample Paths

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.