Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

Abstract

In this article we prove path-by-path uniqueness in the sense of Davie [26] and Shaposhnikov [45] 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.Using this result, we construct weak unique regular solutions in W k,p loc [0, 1] × R d , p > d of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.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 [29], Ambrosio [2] or Crippa-De Lellis [24].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 [25] as well as supremum concentration inequalities.

Noiseless regularisation by noise

“…Proof As before, we can assume s = 0, b smooth; again we decompose T W H b = I (1) + I (2) . Going through the same calculations for I (1) , we obtain…”

Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

“…Butkovsky and Mytnik [6] analysed the regularisation by noise phenomenon for a non-Lipschitz stochastic heat equation and proved path-by-path uniqueness for any initial condition in a certain class of a set of probability one. Amine, Mansouri and Proske [2] investigated path-by-path uniqueness for transport equations driven by fractional Brownian motions with Hurst parameter H < 1/2 and with bounded vector-fields. In [11; 20] the authors solved the regularisation by noise problem from the point of view of additive perturbations.…”

“…More recently, Harang and Perkowski [21] studied the regularisation by noise problem for ODEs with vector fields given by Schwartz distributions and proved that if one perturb such an equation by adding an infinitely regularising path, then it has a unique solution. Kremp and Perkowski [23] looked at multidimensional SDEs with distributional drift driven by symmetric α-stable Lévy processes for α ∈ (1,2]. In all the above mentioned works, the driving noise considered are one parameter processes.…”

Sensitivity Analysis with respect to a Stock Price Model with Rough Volatility via a Bismut-Elworthy-Li Formula for Singular SDEs

“…in the case of a Wiener process and b ∈ L ∞ ([0, T ] × R d ). See also [3] in the case of a fractional Browian motion with Hurst parameter H <…”

“…It turns out that such an estimate-as already mentioned in the Introduction-plays a central role for proving path-by-path uniqueness of solutions (see [13]) to SDE's with additive Wiener or fractional Brownian noise for bounded vector fields b, which is a much stronger property than pathwise uniqueness. See [30], [31] and also [3] for more details.…”