On a coupling of solutions to the interface SDE on a star graph

Inspired by Tsirelson proof of the non Brownian character of WalshBrownian motion ltration on three or more rays, we prove some results on aparticular coupling of solutions to the interface SDE on a star graph, recentlyintroduced. This coupling consists in two solutions which are independentgiven the driving Brownian motion. As a consequence, we deduce that if the stargraph contains 3 or more rays, the argument of the solution at a xed time isindependent of the driving Brownian motion.