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We consider a so-called quantum graph with standard continuity and Kirchhoffvertex conditions where the Kirchhoff vertex condition is perturbed by Gaussiannoise. The strong Feller property of the stochastic parabolic problem isinvestigated by considering the associated Kirchhoff null-controllabilityproblem. We give an abstract characterization of both problems and apply theabstract result to show that the quantum graph setting is very different fromthe classical one dimensional boundary noise setting, where the transitionsemigroup is known to be strong Feller, by giving examples and counterexamplesto the strong Feller property. We also comment on the existence and uniquenessof the invariant measure and the regularity of the solution.

On the strong Feller property of the heat equation on quantum graphs with Kirchoff noise