  We consider a so-called quantum graph with standard continuity and Kirchhoff
vertex conditions where the Kirchhoff vertex condition is perturbed by Gaussian
noise. The strong Feller property of the stochastic parabolic problem is
investigated by considering the associated Kirchhoff null-controllability
problem. We give an abstract characterization of both problems and apply the
abstract result to show that the quantum graph setting is very different from
the classical one dimensional boundary noise setting, where the transition
semigroup is known to be strong Feller, by giving examples and counterexamples
to the strong Feller property. We also comment on the existence and uniqueness
of the invariant measure and the regularity of the solution.
