Averaging along irregular curves and regularisation of ODEs

R. Catellier and M. Gubinelli

Stochastic Processes and their Applications, 2016, vol. 126, issue 8, 2323-2366

Abstract: 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.

Keywords: Regularization by noise; Stochastic differential equation; Young integral; Fractional Brownian motion (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (10)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:126:y:2016:i:8:p:2323-2366

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DOI: 10.1016/j.spa.2016.02.002

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