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- W4324392585 abstract "In this article we prove path-by-path uniqueness in the sense of Davie cite{Davie07} and Shaposhnikov cite{Shaposhnikov16} for SDE's driven by a fractional Brownian motion with a Hurst parameter $Hin(0,frac{1}{2})$, uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.par Using this result, we construct weak unique regular solutions in $W_{loc}^{k,p}left([0,1]timesmathbb{R}^dright)$, $p>d$ of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.par The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lyons cite{DiPernaLions89}, Ambrosio cite{Ambrosio04} or Crippa-De Lellis cite{CrippaDeLellis08}.par Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from cite{DMN92} as well as supremum concentration inequalities. emph{keywords}: Transport equation, Compactness criterion, Singular vector fields, Regularization by noise." @default.
- W4324392585 created "2023-03-16" @default.
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- W4324392585 date "2020-03-13" @default.
- W4324392585 modified "2023-09-28" @default.
- W4324392585 title "Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths" @default.
- W4324392585 doi "https://doi.org/10.48550/arxiv.2003.06200" @default.
- W4324392585 hasPublicationYear "2020" @default.
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