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: 

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 ∇HLr,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.