Three Examples of Brownian Flows on R

We show that the only ow solving the stochastic di erential equation (SDE) on R dXt = 1{Xt>0}W+(dt) + 1{Xt<0}dW−(dt), where W and W− are two independent white noises, is a coalescing ow we will denote φ±. The ow φ± is a Wiener solution. Moreover, K = E[δφ±|W+] is the unique solution (it is also a Wiener solution) of the SDE K s,tf(x) = f(x) + ∫ t s Ks,u(1R+f )(x)W+(du) + 1 2 ∫ t s Ks,uf ′′(x)du for s < t, x ∈ R and f a twice continuously di erentiable function. A third ow φ can be constructed out of the n-point motions of K. This ow is coalescing and its n-point motion is given by the n-point motions of K up to the rst coalescing time, with the condition that when two points meet, they stay together. We note nally that K = E[δφ+|W+].