Averaging along irregular curves and regularisation of ODEs

• Published in: Stochastic processes and their applications Vol. 126; no. 8; pp. 2323 - 2366
• Main Authors: Catellier, R., Gubinelli, M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2016; Elsevier
• Subjects: Fractional Brownian motion; Mathematics; Probability; Regularization by noise; Stochastic differential equation; Young integral; Young integral; Fractional Brownian motion; Regularization by noise; Stochastic differential equation
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2016.02.002

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–Hölder 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.