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

Abstract

We consider a rough differential equation with a non-linear damping drift term: \begin{align*} dY(t) = - |Y|^{m-1} Y(t) dt + σ(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 ${\varphi }^4$ 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.

arXiv comment27 pages, typos fixed and references added

References in corpus (12)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• A theory of regularity structures
• Smooth approximation of stochastic differential equations
• Combinatorics of Rooted Trees and Hopf Algebras
• Runge-Kutta methods and renormalization
• Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations
• A Rough Path Perspective on Renormalization
• Lack of strong completeness for stochastic flows
• Controlling Rough Paths
• Differential equations driven by rough paths: an approach via discrete approximation
• Stability of Deep Neural Networks via discrete rough paths
• Ramification of rough paths

Cited by in corpus (4)

• The structure group for quasi-linear equations via universal enveloping algebras
• Distribution dependent SDEs driven by additive continuous noise
• Renormalised singular stochastic PDEs
• Locality for singular stochastic PDEs