Three examples of Brownian flows on ℝ

We show that the only flow solving the stochastic differential equation (SDE) on R (Formula presented.), where W⁺ and W^- are two independent white noises, is a coalescing flow we will denote by Pdbl^±. tHE FLOW Pdbl^± is aWiener solution of the SDE. Moreover, K⁺ = (Formula presented.) is the unique solution (it is also a Wiener solution) of the SDE (Formula presented) for s < t, x ∈ ℝ and f a twice continuously differentiable function. A third flow Pdbl⁺ can be constructed out of the n-point motions of K⁺ This flow is coalescing and its n-point motion is given by the n-point motions of K⁺ up to the first coalescing time, with the condition that when two points meet, they stay together. We note finally that K⁺ = (Formula presented.).