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We consider a system of stochastic Allen-Cahn equations on a finite networkrepresented by a finite graph. On each edge in the graph a multiplicativeGaussian noise driven stochastic Allen-Cahn equation is given with possiblydifferent potential barrier heights supplemented by a continuity condition anda Kirchhoff-type law in the vertices. Using the semigroup approach forstochastic evolution equations in Banach spaces we obtain existence anduniqueness of solutions with sample paths in the space of continuous functionson the graph. We also prove more precise space-time regularity of the solution.

On the stochastic Allen-Cahn equation on networks with multiplicative noise