M. Gubinelli
The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the Dürr-Connes-Kreimer coproduct on rooted trees. This turns out to be the natural setting for a non-geometric theory of rough paths.
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By using an explicit ordinary differential equation to approximate the exponential solution flow, we extend the universal limit theorem to rough differential equation in Banach space driven by weak geometric rough path, and give the quantitative dependence of solution in term of the initial value, vector field and driving rough path.
We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks, and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integratio…
This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons ("Differential equations driven by rough signals", Revista Mathematica Iber. Vol 14, Nr. 2,215-310, 1998). Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p and the theory of integration of Lip-γone-forms (γ>p-1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and g…
We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-gamma manifolds.
We provide an account for the existence and uniqueness of solutions to rough differential equations under the framework of controlled rough paths. The case when the driving path is $β$-Hölder continuous, for $β>1/3$, is widely available in the literature. In its extension to the case when $β\leqslant1/3,$ a main challenge and missing ingredient is to show that controlled roughs paths are closed under composition with Lipschitz transformations. Establishing such a property precisely, which has a strong algebraic nature, is a main purpose of the present article.
We build a connection between rough path theory and noncommutative algebra, and interpret the integration of geometric rough paths as an example of a non-abelian Young integration. We identify a class of slowly-varying one-forms, and prove that the class is stable under basic operations. In particular rough path theory is extended to allow a natural class of time varying integrands.
We give an overview of the recent approach to the integration of rough paths that reduces the problem to classical Young integration. As an application, we extend an argument of Schwartz to rough differential equations, and prove the existence, uniqueness and continuity of the solution, which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.
Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in "Rough Paths on Manifolds" (Cass, T.…
A Transfer Principle for Branched Rough Paths
本文通过 Kelly 括号扩张建立分支粗糙路径的转移原理，在光滑流形上给出无坐标积分定义，并构造由准几何粗糙路径驱动的流形值粗糙微分方程，将非几何粗糙演算推广到任意低正则性。
We embed the rough integration in a larger geometrical/algebraic framework of integrating one-forms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated paths as integrals of one-forms, and demonstrate that they are stable under basic operations.
1.When equipped with 2-rough norm and restricted to continuous paths with bounded variation, the area operator is a closable unbounded operator. 2.The area defined through Riemann-Stieltjes integral is the only possible candidate to enhance a path with vanishing 2-variation into a geometric 2-rough path. 3.Young integral is extended to p^-1+q^-1=1 by assigning a finer scale continuity.
The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution at each point of the real line in terms of decorated planar forests. For this we develop the notion of planarly branched rough paths, following M. Gubinelli's branched rough paths. The definition is similar to the one in the flat case, the main difference being the replacement of the Butcher--Connes--Kreimer Hopf algebra of non-planar rooted forests by the Munthe-Kaas--Wright Hopf algebra of planar rooted forests. We …
In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly-Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.
Solutions to linear controlled differential equations can be expressed in terms of iterated path integrals of the driving path. This collection of iterated integrals encodes essentially all information about the driving path. While upper bounds for iterated path integrals are well known, lower bounds are much less understood, and it is known only relatively recently that some type of asymptotics for the $n$-th order iterated integral can be used to recover some intrinsic quantitative properties of the path, such as the length of $C^1$ paths. In the present paper, we investigate the simplest ty…
We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of RDEs driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and SDEs on manifolds [É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15] to the …
In this complementary note to [1] (arXiv:1501.05641), we provide an alternative proof for the factorial decay estimate of iterated integrals for geometric rough paths without using the neoclassical inequality. This note intends to aid the readers on the proof in [1] which works also for branched rough paths. Just as in [1], the proof here is an extension of Lyons 94' [4] from Young's integration to geometric rough paths.
In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in Gubinelli (2004). We first show that branched rough paths can equivalently be defined as $γ$-Hölder continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path $\mathbf{X}$ lying above a path $X$, there exists a geometric rough path $\bar{\mathbf{X}}$ lying above an extended path $\bar X$, such that $\b…
We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $Π$-rough paths for $Π=\left( p_{1},\cdots ,p_{k}\right) $, $p_{i}\geq 1$. When there exists $ p\geq 1$ such that $p_{i}=pk_{i}^{-1}\ $for some $k_{i}\in \left\{ 1,\dots , \left[ p\right] \right\} $, we obtain a refined Taylor remainder estimate that contains a factorial decay component. The remainder estimates are in the right order as they are comparable to the next term in the Taylor expansion.
Itô formula for planarly branched rough paths
本文针对包含更多随机过程的平面分支粗糙路径，证明了粗糙度$\frac{1}{4}< α\leq \frac{1}{2}$条件下的Itô公式，扩展了粗糙路径理论。
We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d-dimensional manifold and rough paths on d-dimensional Euclidean space. This last result is a rough path analogue of Cartan's develo…
Rough differential equations and planarly branched universal limit theorem
该研究针对粗糙路径理论中通用极限定理的适用范围问题，利用巴拿赫不动点定理，证明了粗糙度在$\frac{1}{4}<α≤\frac{1}{3}$区间的平面分支粗糙路径的通用极限定理，拓展了该定理的适用场景。
We show in this note how the machinery of C^1-approximate flows devised in the work "Flows driven by rough paths", and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path, can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force in a simple way.
We show in this work how the machinery of C^1-approximate flows introduced in our previous work "Flows driven by rough paths", provides a very efficient tool for proving well-posedness results for path-dependent rough differential equations on flows of the form dϕ= V h(dt) + F X(dt), for smooth enough path-dependent vector fields V,F = (V_1,...,V_\ell), any Holder weak geometric p-rough path X and any a-Holder path h, with a+1/p>1.
We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical derivatives.
Flows driven by multi-indices Rough Paths
本文提出了一种由多索引粗糙路径驱动的粗糙微分方程的解理论，并利用log-ODE方法进行分析。
Recently it was proved that the group of rough paths modulo tree-like equivalence is isomorphic to the corresponding signature group through the signature map S (a generalized notion of taking iterated path integrals). However, the proof of this uniqueness result does not contain any information on how to "see" the trajectory of a (tree-reduced) rough path from its signature, and a constructive understanding on the uniqueness result (in particular on the inverse of S) has become an interesting and important question. The aim of the present paper is to reconstruct a rough path from its signatur…
We prove some results, which are used in arXiv:1406.7871, about weakly geometric rough paths that are well-known in finite dimensions, but need proof in the infinite dimensional setting.
We extend the recently developed rough path theory for Volterra equations from (Harang and Tindel, 2019) to the case of more rough noise and/or more singular Volterra kernels. It was already observed in (Harang and Tindel, 2019) that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of "non-geometric rough paths" d…
We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir (2006) as well as a canonical version of the Itô-Stratonovich correction formula of Hairer-Kelly (2015). Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra. We apply this isomorphism to study signatures of branched rough paths. Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a no…
Rough differential equations and reduced rough paths: a Lie bracket characterization
研究 Hölder 正则性\(\frac13<\alpha\le\frac12\)下粗糙微分方程，从约化粗糙路径角度出发，确定其确定解的障碍，证明在交换向量场情形下反对称 Lévy 面积无关，此为研究粗糙微分方程提供新视角。
Rough path theory
本文综述粗糙路径理论，该理论为低正则项微分方程提供连续性框架，并在随机分析中为不规则路径驱动的随机微分方程赋予新意义，涵盖解的概念、路径签名及应用。
Functional differential equations driven by càdlàg rough paths
本文针对由 càdlàg 粗糙路径驱动的路径依赖粗糙微分方程，建立了解的存在唯一性及 Itô-Lyons 映射的局部 Lipschitz 连续性，并将结果应用于受控、带延迟的 RDEs 等，还构造了对应 Itô 积分的联合粗糙路径。
A theory of differential equations driven by a non-differentiable path has recently been developed by Lyons. We develop an alternative approach to this theory, using (modified Euler approximations), and investigate its applicability to stochastic differential equations driven by Brownian motion. We also give some other examples showing that the main results are reasonably sharp.
We devise in this work a simple mechanism for constructing flows on a Banach space from approximate flows, and show how it can be used in a simple way to reprove from scratch and extend the main existence and well-posedness results for rough differential equations, in the context of dynamics on a Banach space driven by a finite dimensional Holder weak geometric p-rough path, for any 2<p; the explosion question under linear growth conditions on the vector fields, Taylor expansion and convergence rates for Euler estimates are also dealt with. We illustrate our approach by proving an existence an…
Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.
Optimal extension to Sobolev rough paths
该研究证明满足α>1/p>0的Sobolev路径可提升为Lyons意义下的Sobolev粗糙路径，且存在关于严格凸泛函的唯一最优提升，给出了最小范数提升与布朗运动Stratonovich提升的实例，并提及Besov空间推广。
Smooth rough paths, their geometry and algebraic renormalization
本文引入光滑粗糙路径，利用Maurer-Cartan视角实现Lyons延拓定理的纯代数形式及粗糙路径重整化，并扩展至拟几何与Hopf代数框架。
It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in $q$-variation resp. $1/q$-Hölder type metrics on the space of rough paths, for any regularity $1/q \in (0,1]$. We extend this to a new class of Besov-Nikolskii-type metrics, with arbitrary regularity $1/q\in (0,1]$ and integrability $p\in [ q,\infty ]$, where the case $p\in \{ q,\infty \} $ corresponds to the known cases. Interestingly, the result is obtained…
In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of controlled rough one-forms along such a controlled path and their resulting integrals are then defined. This general integration theory does require the introduction of an additional geometric structure on the manifold which we refer to as a "parallelism." The transformation properties of the theory under change of parallelisms is explored. Using these trans…
A rough path approach to pathwise stochastic integration à la Föllmer
本文用粗糙路径扩展 Föllmer 路径积分框架，在二次变差与 Lévy area 存在时统一刻画 Itô、Stratonovich 和后向 Itô 积分，并给出分割不变性的充要条件。
Rough Paths above Weierstrass Functions
针对多元 Weierstrass 函数缺乏显式粗糙路径提升的问题，本文构造其迭代积分，并据此给出由该确定性不规则信号驱动的微分方程解及示例。
The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various ki…
This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon's theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product be…
The geometry of controlled rough paths
本文证明任意阶受控（分支）粗糙路径空间构成连续 Banach 空间场，建立内蕴拓扑与逼近结果，并使粗糙积分及 Itô-Lyons 映射连续且保持结构。
Controlled rough paths: a general Hopf-algebraic setting
本文在涵盖 shuffle、Butcher–Connes–Kreimer 等组合 Hopf 代数的一般框架下建立受控粗糙路径理论，用不动点方法证明提升初值问题解的存在唯一性，并给出关于参数与初值的普适稳健极限定理。
Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total $p$-variation of trained weights for any $p\in[1,3]$. Unlike the $C^1$-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [arXiv:2105.12245]. Mathematically, we interpre…
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