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

By Lisa Beck, Franco Flandoli, Massimiliano Gubinelli, M. Maurelli. Electronic Journal of Probability, 2014

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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.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/a2839f4d1654756d22109e843d1b8310c380cb2b

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Citations in the corpus

Noiseless regularisation by noise (2020, background): “atural choice. Furthermore, a large set of theoretical tools is available to analyse the eect of Brownian perturbations to deterministic evolutions and this topic has a long and extensive literature [7,13,14,21,34,51,52]. Other classes of random perturbations, like fractional Brownian motion (fBm) h”
On the convergence of stochastic transport equations to a deterministic parabolic one (2019, methodology): “hich essentially sharp condition are given by [9], [1]. There is now an extensive literature on the topic of regularization by noise for transport equations, see the review [12] and the references in [4]. However, from the modelling point of view, spaceindependent noise is too simple, since formally”