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

TL;DR

This paper establishes a strong a priori bound for solutions to a rough differential equation with a nonlinear damping term, demonstrating the 'coming down from infinity' property uniformly over initial conditions.

Contribution

It extends algebraic frameworks to derive bounds for rough differential equations with nonlinear damping, including the 'coming down from infinity' property, applicable to arbitrary rough path regularity.

Findings

Proves uniform bounds for solutions regardless of initial data.

Extends algebraic methods for rough paths to nonlinear damping equations.

Demonstrates the 'coming down from infinity' property in this context.

Abstract

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.