A priori bounds for rough differential equations with a non-linear damping term

We consider a rough differential equation with a non-linear damping drift term:dY(t)=−|Y|m−1Y(t)dt+σ(Y(t))dX(t), where m>1, X is a (branched) rough path of arbitrary regularity α>0, and where σ is smooth and satisfies an m and α-dependent growth property. We show a strong a priori bound for Y, which includes the “coming down from infinity” property, i.e. the bound on Y(t) for a fixed t>0 holds uniformly over all choices of initial datum Y(0). The method of proof builds on recent work on a priori bounds for the ϕ4 SPDE in arbitrary subcritical dimension [7]. A key new ingredient is an extension of the algebraic framework which permits to derive an estimate on higher order conditions of a coherent controlled rough path in terms of the regularity condition at lowest level.