On the Fractional Wave Equation

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 / ν . 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