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

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