Unisolvence of unsymmetric random Kansa collocation by Gaussians and other analytic RBF vanishing at infinity

We give a short proof of almost sure invertibility of unsymmetric random Kansa collocation matrices by a class of analytic RBF vanishing at infinity, for the Poisson equation with Dirichlet boundary conditions. Such a class includes popular Positive Definite instances such as Gaussians, Generalized Inverse MultiQuadrics and Matérn RBF. The proof works on general domains in any dimension and with any distribution of boundary collocation points, assuming that the internal collocation points are i.i.d. continuous random variables with respect to any probability density.