  The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field
model of stochastic microscopic dynamics associated with adsorption and
desorption-spin flip mechanisms in the context of surface processes. For such
an equation we consider a multiplicative space-time white noise with diffusion
coefficient of sub-linear growth. Using technics from semigroup theory, we
prove existence, and path regularity of stochastic solution depending on that
of the initial condition. Our results are also valid for the stochastic
Cahn-Hilliard equation with unbounded noise diffusion, for which previous
results were established only in the framework of a bounded diffusion
coefficient. We prove that the path regularity of stochastic solution depends
on that of the initial condition, and are identical to those proved for the
stochastic Cahn-Hilliard equation and a bounded noise diffusion coefficient. If
the initial condition vanishes, they are strictly less than 2-d/2 in space and
1/2-d/8 in time. As expected from the theory of parabolic operators in the
sense of Petrovski, the bi-Laplacian operator seems to be dominant in the
combined model.
