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

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R d ∋ x 7− → φs,t(x) ∈ Rd, s, t ∈ R, for a stochastic differential equation (SDE) of the form dXt = b(t,Xt) dt + dBt, s, t ∈ R, Xs = x ∈ Rd. 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. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant ‘culture ’ in these dynamical systems is that the flow ‘inherits ’ its spatial regularity from that of 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) ◦ dBit = 0, u(0, x) = u0(x), where b is bounded and measurable, u0 is C 1 b and {ei}di=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