Fractional Brownian motion ($\mathbb{E}[B_H(s)B_H(t)]=\frac{1}{2}(s^{2H}+t^{2H}-|s-t{|}^{2H})$)

A fractional Brownian motion with Hurst exponent $H\in (0,1)$ is a centered Gaussian process with the displayed standard covariance function and a continuous modification. Its increment variance is $|s-t{|}^{2H}$. Some sources omit the factor $1/2$, multiplying the process by $\sqrt{2}$; the corresponding increment variance is then $2|s-t{|}^{2H}$. Its canonical pseudometric of a Gaussian process is proportional to $|s-t{|}^H$, which gives finite expected value of its absolute supremum on compact intervals through the Dudley entropy integral.

For fractional Brownian motion, $H\in (0,1)$ controls the variance $|t-s{|}^{2H}$ of increments and the scaling of its finite-dimensional distributions. Larger $H$ means smaller increment variance on intervals shorter than one and greater sample-path Hölder continuity.