This is a joint seminar with the Probability Seminar. Please note special day and time. According to DiPerna-Lions theory, velocity fields with weak derivatives in Lp spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a d-dimensional diffusion with a drift in Lr,q space (r for the spatial variable and q for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that d/r+2/q<1. As an application one show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function H satisfies ∇H∈Lr,q with d/r+2/q<1. As our second application we derive a Constantin-Iyer type circulation formula for certain weak solutions of Navier-Stokes equation.