# Space-time duality for semi-fractional diffusions

## TL;DR

This paper reviews the classical space-time duality for fractional diffusions based on Zolotarev's duality and extends it from stable to semi-stable distributions, broadening the theoretical framework.

## Contribution

It generalizes the existing space-time duality for fractional diffusions from stable to semi-stable distributions, providing a broader theoretical foundation.

## Key findings

- Revisits Zolotarev's duality for stable densities.
- Extends space-time duality to semi-stable distributions.
- Provides a generalized framework for fractional diffusions.

## Abstract

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 $\frac1{\alpha}$-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 $\frac1{\alpha}$. We review on this space-time duality and take it as a recipe for a generalization from the stable to the semistable situation.