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 ∈ (1, 2) to the density of a 1 α -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 timefractional differential equations of order 1 α . We review on this space-time duality and take it as a recipe for a generalization from the stable to the semistable situation.