Space-time duality for semi-fractional diffusions

Almost sixty years ago Zolotarev proved a duality result which relates an $\alpha$-stable density for $\alpha \in (1,2)$ to the density of a $\frac{1}{\alpha }$-stable distribution on the positive real line. In recent years Zolotarev duality was the key to show space-time duality for fractional diffusions stating that certain heat-type fractional equations with a fractional derivative of order $\alpha$ in space are equivalent to corresponding time-fractional differential equations of order $\frac{1}{\alpha }$. We review on this space-time duality and take it as a recipe for a generalization from the stable to the semistable situation.