Isotropic stochastic flow of homeomorphisms on RdRd associated with the critical Sobolev exponent

We consider the critical Sobolev isotropic Brownian flow in Rd(d≥2). On the basis of the work of LeJan and Raimond [Y. LeJan, O. Raimond, Integration of Brownian vector fields, Ann. Probab. 30 (2002) 826–873], we prove that the corresponding flow is a flow of homeomorphisms. As an application, we construct an explicit solution, which is also unique in a certain space, to the stochastic transport equation when the associated Gaussian vector fields are divergence free.