We survey and study the regularity of the sample paths of Brownian motion and fractional Brownian motion. They are the examples of the continuous, but nowhere differentiable Gaussian processes.
We focus on the H？lder regularity of the sample paths. In the case of Brownian motion the H？lder exponent is less than 1/2. For fractional Brownian motion of Hurst parameter H the sample paths has the H？lder exponent is less than H. These follow from the Kolmogorov continuity criterion. On the other hand, the law of the iterated logarithm for Gaussian processes shows that the sample paths of Brownian motion and fractional Brownian motion are nowhere H？lder continuous when the H？lder exponent is greater than equal to 1/2 and H, respectively.
