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

TL;DR

This paper investigates a special coupling of solutions to the interface stochastic differential equation on a star graph, revealing independence properties of the solution's argument from the driving Brownian motion when the graph has three or more rays.

Contribution

It introduces a novel coupling of solutions to the interface SDE on star graphs and analyzes their independence properties, extending understanding of WalshBrownian motion characteristics.

Findings

The coupling involves two solutions independent given the Brownian motion.

If the star graph has three or more rays, the solution's argument at a fixed time is independent of the Brownian motion.

The results are inspired by Tsirelson's proof related to Walsh-Brownian motion.

Abstract

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.