Averaging along irregular curves and regularisation of ODEs

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Catellier, R.^1,2  Gubinelli, M.^1,2,3 

[1] Univ Paris 09, CEREMADE, UMR 7534, F-75775 Paris 16, France

[2] CNRS, F-75700 Paris, France

[3] Inst Univ France, Paris, France

关键词: Regularization by noise;    Stochastic differential equation;    Young integral;    Fractional Brownian motion;   

DOI  :  10.1016/j.spa.2016.02.002

来源: Elsevier

【 摘 要 】

We consider the ordinary differential equation (ODE) dx(t) = b(t, x(t))d(t) + dw(t) 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 p-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 is an element of (0, 1), we prove that almost surely the ODE admits a solution for all b in the Besov-Holder space B-infinity,infinity(alpha+1), with alpha > -1/2H. If alpha > 1 - 1/2H then the solution is unique among a natural set of continuous solutions. If H > 1/3 and alpha > 3/2 - 1/2H or if alpha > 2 - 1/2H then the equation admits a unique Lipschitz flow. Note that when alpha < 0 the vector field b is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

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