Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness
Maurelli, Mario; Gubinelli, Massimiliano; Flandoli, Franco; Beck, Lisa (2014), Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness. https://basepub.dauphine.fr/handle/123456789/12506
TypeDocument de travail / Working paper
External document linkhttp://arxiv.org/abs/1401.1530
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Abstract (EN)Linear stochastic transport and continuity equations with drift in critical Lp spaces are considered. A result of Sobolev regularity of solutions is proved, false for the corresponding deterministic equations. Thus noise prevents shocks for transport equation and singularities in the density for continuity equation, starting from smooth initial conditions. The technique needed to reach the critical case is new and based on parabolic equations satis ed by moments of rst derivatives of the solution, opposite to previous works based on stochastic ows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, the 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 di erential equation (sDE) (roughly speaking, existence and uniqueness of ows and their C regularity, strong uniqueness for the sDE when the initial datum has di use 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.
Subjects / Keywordspartial differential equation (PDE); stochastic differential equation (SDE)
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