Seminar

According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ 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 $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $d/r+2/q<1$. This is also true when $d/r+2/q=1$ if we replace $L^{r,q}$ with an appropriate Lorenz space.

As an application one can show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ 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. I also discuss the connection between the above results and the Hofer Geometry/Symplectic Rigidity.