Weighted Dirichlet spaces that are de Branges-Rovnyak spaces with equivalent norms

For a positive Borel measure μ in the unit disc, we examine which weighted Dirichlet spaces $D_{\mu }$ can be identified with de Branges-Rovnyak spaces $H_b$, with equivalent norms. We prove a necessary condition for the equality $D_{\mu }=H_b$ and we explore its consequences. For Carleson measures μ, we give a necessary and sufficient condition for the equality $D_{\mu }=H_{b_{\mu }}$, for a certain outer function $b_{\mu }$ related with the balayage of μ on the unit circle, and we provide examples of those spaces.