Three examples of Brownian flows on \RR

2011/11/08 by Jan, Yves Le, Raimond, Olivier
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1111.1846

Abstract

We show that the only flow solving the stochastic differential equation (SDE) on \RR dX_t = 1_\X_tgt;0\W₊(dt) + 1_\X_tlt;0\dW_-(dt), where W⁺ and W^- are two independent white noises, is a coalescing flow we will denote \p^±. The flow \p^± is a Wiener solution. Moreover, K⁺=\E[δ_\p^±|W₊] is the unique solution (it is also a Wiener solution) of the SDE K⁺_s,tf(x)=f(x)+∫_s^t K_s,u(1_\RR⁺f')(x)W₊(du)+(1/2) ∫_s^t K_s,uf"(x) du for s

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