Electronic Journal of Probability

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

Lisa Beck, Franco Flandoli, Massimiliano Gubinelli, and Mario Maurelli

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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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Electron. J. Probab., Volume 24 (2019), paper no. 136, 72 pp.

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

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Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness 35B65: Smoothness and regularity of solutions

stochastic transport equation stochastic continuity equation regularization by noise regularity path-by-path uniqueness

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Beck, Lisa; Flandoli, Franco; Gubinelli, Massimiliano; Maurelli, Mario. Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness. Electron. J. Probab. 24 (2019), paper no. 136, 72 pp. doi:10.1214/19-EJP379. https://projecteuclid.org/euclid.ejp/1574996477

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