From snapping-out Brownian motions to Walsh’s spider processes on star-like graphs

We prove that a snapping-out Brownian motion with large permeability coefficients is a good approximation of Walsh’s spider process on the star-like graph K_{1,k} . Thus, the latter process can be seen as a Brownian motion perturbed by a trace of semi-permeable membrane at the graph’s center. Besides convergence of processes and semigroups we establish, via the Lord Kelvin method of images and analysis of ergodic properties of matrices involved, convergence of cosine families underlying the semigroups and thus gain additional insight into the approximation theorem.