We consider a Hamiltonian lattice field model with two conserved quantities,
energy and volume, perturbed by stochastic noise preserving the two previous
quantities. It is known that this model displays anomalous diffusion of energy
of fractional type due to the conservation of the volume [5, 3]. We superpose
to this system a second stochastic noise conserving energy but not volume. If
the intensity of this noise is of order one, normal diffusion of energy is
restored while it is without effect if intensity is sufficiently small. In this
paper we investigate the nature of the energy fluctuations for a critical value
of the intensity. We show that the latter are described by an
Ornstein-Uhlenbeck process driven by a L\'evy process which interpolates
between Brownian motion and the maximally asymmetric 3/2-stable L\'evy process.
This result extends and solves a problem left open in [4].Comment: to appear in AIHP
