Sobolev Differentiable Flows for Singular Stochastic Differential Equations
Tuesday, November 20, 2012 - 10:30am - 12:00pm
We show the existence of a unique stochastic flow of Sobolev diffeomorphisms for stochastic differential equations (SDEs) with bounded measurable drift coefficients. This result is counter-intuitive: The dominant 'culture' in dynamical systems is that the flow 'inherits' its spatial regularity from the driving vector fields. Spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the stochastic transport equation with singular coefficients (cf. work by Kunita (1990); and Flandoli-Gubinelli-Priola (2010)). The corresponding deterministic transport equation does not in general have a solution (Ambrosio (2004)). If time permits, we will construct a Sobolev differentiable stochastic flow of diffeomorphisms for one-dimensional SDEs driven by bounded measurable diffusion coefficients. No uniqueness of solutions to the SDE is presumed!