Averaging along irregular curves and regularisation of ODEs

By Rémi Catellier, Massimiliano Gubinelli. 2012

181 citations recorded by the source corpus.

We consider the ordinary differential equation (ODE) dxt=b(t,xt)dt+dwt where w is a continuous driving function and b is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path w on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of ρ-irregularity and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function w sampled according to the law of the fractional Brownian motion of Hurst index H∈(0,1), we prove that almost surely the ODE admits a solution for all b in the Besov–Holder space B∞,∞α+1 with α>−1/2H. If α>1−1/2H then the solution is unique among a natural set of continuous solutions. If H>1/3 and α>3/2−1/2H or if α>2−1/2H then the equation admits a unique Lipschitz flow. Note that when α<0 the vector field b is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/781d80777124b238cf00e824be4223af8af72087

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Citations in the corpus

Quantifying a convergence theorem of Gyöngy and Krylov (2021, methodology): “Recently, in [7] a new approach was introduced to obtain such bounds, based on the sewing methods inspired by [8, 26].”
Noiseless regularisation by noise (2020, methodology): “This limit is called in [10] a non-linear Young integral and denoted as ∫ t”
Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, methodology): “Let us mention that, in some cases, it is possible to have regularization by noise even for random drifts, see [18] and related work, assuming a suitable Hölder continuity of the drift, or [31, 69], assuming Malliavin differentiability of the drift.”
Rough paths and 1d SDE with a time dependent distributional drift: application to polymers (2014, methodology): “We start with reminders about rough paths, following Gubinelli’s approach in [17].”
Distribution dependent SDEs driven by additive fractional Brownian motion (2021, methodology): “Treating DDSDEs driven by fBm thus requires a novel set of tools and ideas; our strategy in this paper builds on the work of Catellier and Gubinelli [9], which represented a major breakthrough in the study of standard SDEs driven by fBm of the form”