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We consider a reaction-diffusion equation on a network subjected to dynamicboundary conditions, with time delayed behaviour, also allowing formultiplicative Gaussian noise perturbations. Exploiting semigroup theory, werewrite the aforementioned stochastic problem as an abstract stochastic partialdifferential equation taking values in a suitable product Hilbert space, forwhich we prove the existence and uniqueness of a mild solution. Eventually, astochastic optimal control application is studied.

Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions