2020/03/13 by Oussama Amine, Amine, Oussama, Abdol-Reza Mansouri +3
Economics, Econometrics and Finance · Mathematics · #49N60 #60H10 #91G80 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2003.06200
openalex publication_date 2020/03/13 · openalex created_date 2023/03/16 · openalex updated_date 2026/07/28
In this article we prove path-by-path uniqueness in the sense of Davie \citeDavie07 and Shaposhnikov \citeShaposhnikov16 for SDE's driven by a fractional Brownian motion with a Hurst parameter H∈(0,(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 Wlock,p([0,1]×ℝd), 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 \citeDiPernaLions89, Ambrosio \citeAmbrosio04 or Crippa-De Lellis \citeCrippaDeLellis08.\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 \citeDMN92 as well as supremum concentration inequalities. keywords: Transport equation, Compactness criterion, Singular vector fields, Regularization by noise.