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.
Highlights
In this paper we study in detail the solution of the time-fractional equation d
We first prove that the Fourier transform of the solution of the Cauchy problem (1) and (2) is
For Γ uniform on the upper and lower hemispheres of Sd−1 = {s ∈ Rd : ksk = 1}, we prove that (8) yields the characteristic functions in square brackets of Formula (7)
Summary
We first prove that the Fourier transform of the solution of the Cauchy problem (1) and (2) is. 1 < ν < 2 the inversion of (7) is presented in [3] with the conclusion that the solution of (1) is the distribution of a stable symmetric process of order 2/ν. In this note we present some relationship between stable processes (and their inverses) with fractional equations. Some simple and well known results state that a symmetric stable process Sα (t), 0 < α ≤ 2 with characteristic function. For the d-dimensional isotropic stable process Sα (t) characteristic function, α d. Some details about time-fractional derivatives can be found in [10]. For 0 < ν < 1 the solution of (18) corresponds to the distribution of the vector process.
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