I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not?
Many thanks.
2026-03-27 16:02:29.1774627349
Sample continuity of Brownian motion
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The paths of a fractional Brownian motion are (a.s.) locally Hölder continuous of any order $\gamma<H$ where $H$ denotes the Hurst index. In particular, the sample paths are continuous. The Hölder continuity follows from the Kolmogorov-Chentsov theorem, see this question.