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Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

We consider the long time, large scale behavior of the Wigner transform $W_\eps(t,x,k)$ of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $γ_0>0$ such that for any $γ\in(0,γ_0]$ the weak limit of $W_\eps(t/\eps^{3/2γ},x/\eps^γ,k)$, as $\eps\ll1$, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ with $\hat c>0$. In the pinned case an analogous result can be claimed for $W_\eps(t/\eps^{2γ},x/\eps^γ,k)$ but the limit satisfies then the usual heat equation.