带非线性阻尼项的粗糙微分方程的先验界
A priori bounds for rough differential equations with a non-linear damping term
摘要：
考虑一类带非线性阻尼漂移项的粗糙微分方程：\begin{align*} dY(t) = - |Y|^{m-1} Y(t) dt + σ(Y(t)) dX(t), \end{align*}其中$X$是具有任意正则性的分支粗糙路径$α>0$，$m>1$，其中$σ$是光滑的并且满足$m$和$α$依赖的增长性质.我们展示了一个强的先验界$Y$，其中包括“从无穷大下来”的属性，即对$Y(t)$上的约束为固定的$t>0$保持一致的初始数据$Y(0)$的所有选择。的方法证明建立在最近的工作由钱德拉，Moinat和韦伯的先验界的$ϕ^4$ SPDE在任意次临界尺寸。一个关键的新的成分是一个扩展的代数框架，它允许推导出一个估计的高阶条件的一致性控制粗糙路径的规则性条件在最低水平。
英文摘要：
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 $α>0$, $m>1$ 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 by Chandra, Moinat and Weber on a priori bounds for the $ϕ^4$ 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.
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