  Super-diffusion, characterized by a spreading rate $t^{1/\alpha}$ of the
probability density function $p(x,t) = t^{-1/\alpha} p \left( t^{-1/\alpha} x ,
1 \right)$, where $t$ is time, may be modeled by space-fractional diffusion
equations with order $1 < \alpha < 2$. Some applications in biophysics (calcium
spark diffusion), image processing, and computational fluid dynamics utilize
integer-order and fractional-order exponents beyond than this range ($\alpha >
2$), known as high-order diffusion, or hyperdiffusion. Recently, space-time
duality, motivated by Zolotarev's duality law for stable densities, established
a link between time-fractional and space-fractional diffusion for $1 < \alpha
\leq 2$. This paper extends space-time duality to fractional exponents
$1<\alpha \leq 3$, and several applications are presented. In particular, it
will be shown that space-fractional diffusion equations with order $2<\alpha
\leq 3$ model sub-diffusion and have a stochastic interpretation. A space-time
duality for tempered fractional equations, which models transient anomalous
diffusion, is also developed.
