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: \begin{align*} dY(t) = - |Y|^{m-1} Y(t) dt + \sigma(Y(t)) dX(t), \end{align*} where $X$ is a branched rough path of arbitrary regularity $\alpha >0$, $m>1$ and where $\sigma$ is smooth and satisfies an $m$ and $\alpha$-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 by Chandra, Moinat and Weber on a priori bounds for the $\phi^4$ SPDE in arbitrary subcritical dimension. 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.