Three examples of Brownian flows on $\RR$

TL;DR

This paper characterizes the unique coalescing flow solving a specific SDE driven by two independent white noises on the real line, and explores its properties and solutions.

Contribution

It proves the uniqueness of the flow solving the SDE with two independent white noises and describes its Wiener solution and related equations.

Findings

The flow solving the SDE is unique and coalescing.

The flow is a Wiener solution.

The measure $K^+$ is the unique solution to a related SDE.

Abstract

We show that the only flow solving the stochastic differential equation (SDE) on $\RR$ $$dX_t = 1_{\{X_t>0\}}W_+(dt) + 1_{\{X_t<0\}}dW_-(dt),$$ where $W^+$ and $W^-$ are two independent white noises, is a coalescing flow we will denote $\p^{\pm}$. The flow $\p^\pm$ is a Wiener solution. Moreover, $K^+=\E[\delta_{\p^\pm}|W_+]$ is the unique solution (it is also a Wiener solution) of the SDE $$K^+_{s,t}f(x)=f(x)+\int_s^t K_{s,u}(1_{\RR^+}f')(x)W_+(du)+(1/2) \int_s^t K_{s,u}f"(x) du$$ for $s