Correlated coalescing Brownian flows on R and the circle

Abstract

We consider a stochastic differential equation on the real line which is driven by two correlated Brownian motions B+ and B- respectively on the positive half line and the negative half line. We assume |d〈B+,B-〉t| ≤ ρ dt with ρ ∈ [0, 1). We prove it has a unique flow solution. Then, we generalize this flow to a flow on the circle, which represents an oriented graph with two edges and two vertices. We prove that both flows are coalescing. Coalescence leads to the study of a correlated reflected Brownian motion on the quadrant. Moreover, we find the distribution of the hitting time to the origin of a reflected Brownian motion. This has implications for the effect of the correlation coefficient ρ on the coalescence time of our flows.