*Article* **On the Fractional Wave Equation**

#### **Francesco Iafrate and Enzo Orsingher \***

Dipartimento di Scienze Statistiche, Sapienza, University of Rome, 00185 Rome, Italy; francesco.iafrate@uniroma1.it

**\*** Correspondence: enzo.orsingher@uniroma1.it

Received: 13 May 2020; Accepted: 27 May 2020; Published: 31 May 2020

**Abstract:** In this paper we study the time-fractional wave equation of order 1 < *ν* < 2 and give a probabilistic interpretation of its solution. In the case 0 < *ν* < 1, *d* = 1, the solution can be interpreted as a time-changed Brownian motion, while for 1 < *ν* < 2 it coincides with the density of a symmetric stable process of order 2/*<sup>ν</sup>*. We give here an interpretation of the fractional wave equation for *d* > 1 in terms of laws of stable *d*−dimensional processes. We give a hint at the case of a fractional wave equation for *ν* > 2 and also at space-time fractional wave equations.

**Keywords:** Hankel contours; multivariate stable processes; contour integrals; fractional laplacian
