Gaussian estimates on networks with dynamic stochastic boundary conditions

In this paper we prove the existence and uniqueness for the solution to a stochastic reaction-diffusion equation, defined on a network, and subjected to nonlocal dynamic stochastic boundary conditions. The result is obtained by deriving a Gaussian-type estimate for the related leading semigroup, under rather mild regularity assumptions on the coefficients. An application of the latter to a stochastic optimal control problem on graphs, is also provided.