Open Access
2019 Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness
Lisa Beck, Franco Flandoli, Massimiliano Gubinelli, Mario Maurelli
Electron. J. Probab. 24: 1-72 (2019). DOI: 10.1214/19-EJP379


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.


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Lisa Beck. Franco Flandoli. Massimiliano Gubinelli. Mario Maurelli. "Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness." Electron. J. Probab. 24 1 - 72, 2019.


Received: 27 August 2018; Accepted: 20 October 2019; Published: 2019
First available in Project Euclid: 29 November 2019

zbMATH: 07142930
MathSciNet: MR4040996
Digital Object Identifier: 10.1214/19-EJP379

Primary: 60H10 , 60H15
Secondary: 35A02 , 35B65

Keywords: Path-by-path uniqueness , regularity , Regularization by noise , stochastic continuity equation , stochastic transport equation

Vol.24 • 2019
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