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

We consider a reaction-diffusion equation on a network subjected to dynamic boundary conditions, with time delayed behavior, also allowing for multiplicative Gaussian noise perturbations. Exploiting semigroup theory, we rewrite the aforementioned stochastic problem as an abstract stochastic partial differential equation taking values in a suitable product Hilbert space, for which we prove the existence and uniqueness of a mild solution. Eventually, a stochastic optimal control application is studied.