  We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion
equations which are additively perturbed by a stochastic forcing (time white
noise). The conclusion is that the long time, large space behavior of the
solutions is governed by an interface moving with curvature dependent normal
velocity which is additively perturbed by time white noise. The result is
global in time and does not require any regularity assumptions on the evolving
front. The main tools are (i)~the notion of stochastic (pathwise) solution for
nonlinear degenerate parabolic equations with multiplicative rough (stochastic)
time dependence, which has been developed by the authors, and (ii)~the theory
of generalized front propagation put forward by the second author and
collaborators to establish the onset of moving fronts in the asymptotics of
reaction-diffusion equations.
