**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 *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. 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*∈*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.

# Diffusions with Rough Drifts and Stochastic Symplectic Maps

Speaker:

Fraydoun Rezakhanlou,UC Berkeley

Date:

Mar 4 2015 - 3:00pm

Event type:

Analysis of Fluids and Related Topics

Room:

Fine 322

Abstract: