Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

We prove the uniqueness of self-similar $D_4$-symmetric resistance forms on unconstrained Sierpinski carpets ($\mathcal{U}\mathcal{S}\mathcal{C}$'s). Moreover, on a sequence of $\mathcal{U}\mathcal{S}\mathcal{C}$'s $K_n,n\ge 1$ converging in Hausdorff metric, we show that the associated diffusion processes converge in distribution if and only if the geodesic metrics on $K_n,n\ge 1$ are equicontinuous with respect to the Euclidean metric.