Three examples of Brownian flows on $\RR$

Abstract

We show that the only flow solving the stochastic differential equation (SDE) on $\RR$ $$d{X}_t=1_{\{X_t>0\}}{W}_{+}(dt)+1_{\{X_t<0\}}d{W}_{-}(dt),$$ 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, $K^+=\E[δ_{\p^\pm}|W_+]$ 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

32 pages

References in corpus (1)

• The Brownian web: Characterization and convergence