  In [2] it has been proved that a linear Hamiltonian lattice field perturbed
by a conservative stochastic noise belongs to the 3/2-L\'evy/Diffusive
universality class in the nonlinear fluctuating theory terminology [15], i.e.
energy superdiffuses like an asymmetric stable 3/2-L\'evy process and volume
like a Brownian motion. According to this theory this should remain valid at
zero tension if the harmonic potential is replaced by an even potential. In
this work we consider a quartic anharmonicity and show that the result obtained
in the harmonic case persists up to some small critical value of the
anharmonicity.
