Resumen de Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

Salah-Eldin A. Mohammed, Torstein K Nilssen, Frank Proske

• In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms Rd∋x⟼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;W1,p(Rd,w)) for all s,t and all p∈(1,∞), where W1,p(Rd,w) denotes a weighted Sobolev space with weight w possessing a pth 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+∑di=1ei⋅Du(t,x)∘dBit=0,u(0,x)=u0(x), where b is bounded and measurable, u0 is C1b and {ei}di=1 a basis for Rd. It is well known that the deterministic counterpart of the above equation does not in general have a solution.