Coalescing property of an interface SDE with correlated noises

When a tuple of Wiener noises, not necessarily independent, is indexed by rays of a weighted metric star graph, one can consider the coupling of Walsh Brownian motions so that they move on each ray parallel to the corresponding noises. A stochastic differential equation describing such a system is called an interface stochastic differential equation. Extending the results of Hajri and Raimond, and Çağlar, Hajri and Karakuş, we determine the condition for the coalescence of the two-point motion driven by a common interface SDE associated with possibly correlated Wiener noises. To show this, we investigate the explicit representation of the continuous-time Markov additive process derived from the Lamperti transformation of the star-graph-valued self-similar Markov process, which is the time-changed two-point motion by skipping the time at which both are away from the origin. We also show that this self-similar process satisfies Cramér’s condition with the Cramér number less than the self-similarity index 1 if and only if the essential number of rays is more than 2.