Sobolev Differentiable Stochastic Flows for SDE’s with Singular Coefficients: Applications to the Transport Equation

dXt = b(t,Xt) dt + dBt, s, t ∈ R, Xs = x ∈ R. The above SDE is driven by a bounded measurable drift coefficient b : R × Rd → Rd and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow φs,t(·) of the SDE lives in the space L2(Ω;W 1,p(Rd, w)) for all s, t and all p > 1, where W 1,p(Rd, w) denotes a weighted Sobolev space with weight w possessing a p-th moment with respect to Lebesgue measure on Rd. This result is counter-intuitive, since the dominant ‘culture’ in stochastic (and deterministic) dynamical systems is that the flow ‘inherits’ its spatial regularity from the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation { dtu(t, x)+ (b(t, x) ·Du(t, x))dt + ∑d i=1 ei ·Du(t, x) ◦ dBi t = 0, u(0, x) = u0(x), where b is bounded and measurable, u0 is C 1 b and {ei}i=1 a basis for Rd. It is well-known that the deterministic counter part of the above equation does not in general have a solution. Using stochastic perturbations and our analysis of the above SDE, we establish a deterministic flow of Sobolev diffeomorphisms for classical one-dimensional (deterministic) ODE’s driven by discontinuous vector fields. Furthermore, and as a corollary of the latter result, we construct a Sobolev stochastic flow of diffeomorphisms for one-dimensional SDE’s driven by discontinuous diffusion coefficients.