A kinetic model approximation of Walsh's spider process on the infinite star-like graph

We consider processes of deterministic motions on $k$ copies of the star-like graph $S_k= K_{1,k}$ with $k$ edges which are perturbed by two stochastic mechanisms: one caused by interfaces located at the graphs' centers, the other describing jumps between different copies of the same edge. We prove that diffusing scaling of these processes leads in the limit to the Walsh's spider process on $S_k$.