In this talk, the Navier-Stokes equation on R^d (d > 3) formulated on Besov spaces is considered. Using a stochastic forward-backward differential system, the local existence of a unique solution in B_{p,p}^r, with
r > 1 + d/p is obtained. We also show the convergence to solution of the Euler equation when the viscosity tends to zero. Moreover, we prove the local existence of a unique solution in B_{p,p}^r, with p > 1, 1 < q < 1,
r > max(1; d/p ); here the maximal time interval depends on the viscosity .The content of the paper is based on a joint paper with Ana Bela Cruzeiro and Zhong-Min Qian.