  We consider a system of stochastic Allen-Cahn equations on a finite network
represented by a finite graph. On each edge in the graph a multiplicative
Gaussian noise driven stochastic Allen-Cahn equation is given with possibly
different potential barrier heights supplemented by a continuity condition and
a Kirchhoff-type law in the vertices. Using the semigroup approach for
stochastic evolution equations in Banach spaces we obtain existence and
uniqueness of solutions with sample paths in the space of continuous functions
on the graph. We also prove more precise space-time regularity of the solution.
