Fractional Brownian motion seems fairly a straight forward random process with a kind of auto-correlation function,
$$ \mathbb{E}\left[ B^H_t B^H_s \right] = \frac{1}{2} \left( |t|^{2H} + |s|^{2H} - |t-s|^{2H} \right) $$
But this can easily be derived like,
$$\begin{align} f(|t - s|) &= \mathbb{E}\left[ (B_t - B_s)^2 \right] \\ &= \mathbb{E}\left[ (B_t)^2 + (B_s)^2 - 2B_tB_s \right] \\ &= f(|t|) + f(|s|) - 2 \mathbb{E}\left[ B_tB_s \right] \end{align}$$
taking $ B_0 = 0 $. And then letting $ f(|t - s|) = | t - s |^{2H} $.
As such, there seems nothing special per say about fractional Brownian motion and using the Hurst exponent.
Are other fractional brownian motion forms studied? For example, ones with changing long/short scale correlations?
