受控路径的泛函表示与泛函演算
Functional representation and functional calculus for controlled paths
AI总结：
本文建立非预期泛函演算与受控路径相容族间的联系，证明逆表示定理：相容高阶受控Taylor估计的系数族可表示为基泛函的迭代垂直导数，并应用于路径依赖粗糙微分方程的适定性。
AI中文摘要：
我们研究了非预期泛函演算与受控路径的相容族之间的关系。对于满足水平Lipschitz正则性的非预期泛函，我们证明了迭代垂直导数生成沿Hölder控制的相容高阶受控Taylor展开，且余项指数依赖于水平。我们从这些估计中推导出一个粗糙积分准则，并证明对于满足$\tfrac{1}{2}\geq \gamma>\sqrt{2}-1$的$\gamma$-Hölder控制，一阶余项估计得以恢复。我们的主要结果是一个逆表示定理。我们考虑沿$\gamma$-Hölder路径满足相容高阶受控Taylor估计的非预期泛函$G_0,\ldots,G_p$。在自然连续性和相容性假设下，我们证明它们可以表示为基泛函的迭代垂直导数：$ G_j=\nabla_\omega^jG_0,\qquad j=1,\ldots,p.$ 因此，相容受控族的Gubinelli系数是对称的，并且由其基泛函唯一确定；特别地，Gubinelli导数被识别为泛函Itô演算中引入的垂直导数，从而赋予系数层级以内在的路径空间微分结构。我们证明了这类相容系数族在允许的非预期泛函复合下是稳定的，并推导出相应的泛函链式法则。作为应用，我们获得了一类具有Volterra记忆的路径依赖粗糙微分方程的适定性，并识别了所得路径依赖粗糙系数的Gubinelli导数。
英文摘要：
We study the relation between non-anticipative functional calculus and compatible families of controlled paths. For a non-anticipative functional satisfying horizontal Lipschitz regularity, we show that the iterated vertical derivatives generate compatible higher-order controlled Taylor expansions along Hölder controls, with level-dependent remainder exponents. We derive a rough-integration criterion from these estimates and show that, for $γ-$Hölder controls with $\tfrac{1}{2}\geq γ>\sqrt{2}-1$, the first-order remainder estimate is recovered. Our main result is a converse representation theorem. We consider non-anticipative functionals $G_0,\ldots,G_p$ which satisfy compatible higher-order controlled Taylor estimates along $γ$-Hölder paths. Under natural continuity and compatibility assumptions, we prove that they can be represented as the iterated vertical derivatives of the base functional: $ G_j=\nabla_ω^jG_0,\qquad j=1,\ldots,p.$ Thus the Gubinelli coefficients of a compatible controlled family are symmetric and uniquely determined by its base functional; in particular, the Gubinelli derivative is identified with the vertical derivative introduced in functional Itô calculus, giving the coefficient hierarchy an intrinsic path-space differential structure. We show that this class of compatible coefficient families is stable under admissible non-anticipative functional composition and derive the corresponding functional chain rule. As an application, we obtain well-posedness for a class of path-dependent rough differential equations with Volterra memory, and identify the Gubinelli derivative of the resulting path-dependent rough coefficient.
