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Super-diffusion, characterized by a spreading rate $t^{1/\alpha}$ of theprobability 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 diffusionequations with order $1 < \alpha < 2$. Some applications in biophysics (calciumspark diffusion), image processing, and computational fluid dynamics utilizeinteger-order and fractional-order exponents beyond than this range ($\alpha >2$), known as high-order diffusion, or hyperdiffusion. Recently, space-timeduality, motivated by Zolotarev's duality law for stable densities, establisheda 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, itwill be shown that space-fractional diffusion equations with order $2<\alpha\leq 3$ model sub-diffusion and have a stochastic interpretation. A space-timeduality for tempered fractional equations, which models transient anomalousdiffusion, is also developed.

Space-Time Duality and High-Order Fractional Diffusion