  We consider one-dimensional infinite chains of harmonic oscillators with
stochastic perturbations and long-range interactions which have polynomial
decay rate $|x|^{-\theta}, x \to \infty, \theta > 1$, where $x \in \mathbb{Z}$
is the interaction range. We prove that if $2< \theta \le 3$, then the time
evolution of the macroscopic thermal energy distribution is superdiffusive and
governed by a fractional diffusion equation with exponent $\frac{3}{7-\theta}$,
while if $\theta > 3$, then the exponent is $\frac{3}{4}$. The threshold is $
\theta = 3$ because the derivative of the dispersion relation diverges as $k
\to 0$ when $\theta \le 3$.
