Space-time duality for semi-fractional diffusions

Abstract

Almost sixty years ago Zolotarev proved a duality result which relates an $α$-stable density for $α\in(1,2)$ to the density of a $\frac1α$-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 $α$ in space are equivalent to corresponding time-fractional differential equations of order $\frac1α$. We review on this space-time duality and take it as a recipe for a generalization from the stable to the semistable situation.