Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

Lisa Beck,Massimiliano Gubinelli,Franco Flandoli,Mario Maurelli

doi:10.1214/19-ejp379

Abstract

In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their $C^\alpha$ regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.

Highlights

Introduction to the ejpecp Class1 IntroductionLet b : [0, T ] × Rd → Rd, for d ∈ N, be a deterministic, time-dependent vector field, that we call drift
Let us mention that the approach to uniqueness of [4] shares some technical steps with the results described in Section 1.6: renormalization of solutions, Itô reformulation of the Stratonovich equation and expected value
There are some results on weak well-posedness for measure-valued drifts, see [9], and distribution-valued drifts, see [44, 28, 15], but it is unclear whether they apply to the limit case p = d: for example, the result in [9], when restricted to measures with density b with respect to the Lebesgue measure, requires p > d, see [9, Example 2.3]

Summary

Introduction

Let b : [0, T ] × Rd → Rd, for d ∈ N, be a deterministic, time-dependent vector field, that we call drift. Let (Wt)t≥0 be a Brownian motion in Rd, defined on a probability space (Ω, A, P ) with respect to a filtration (Gt)t≥0 and let σ be a real number. The following three stochastic equations are (at least formally) related: 1. DX = b(t, X)dt + σdWt, X0 = x, where x ∈ Rd; the unknown (Xt)t∈[0,T ] is a stochastic process in Rd; 2. The stochastic transport equation (sTE ) (sDE). Du + b · ∇udt + σ∇u ◦ dWt = 0, u|t=0 = u0, (sTE). Where u0 : Rd → R, b · ∇u = d i=1 bi∂xi u, ∇u ◦ dWt = d i=1

Objectives

Results

Conclusion