Yet another introduction to rough paths. (English) Zbl 1198.60002

Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLII. Berlin: Springer (ISBN 978-3-642-01762-9/pbk; 978-3-642-01763-6/ebook). Lecture Notes in Mathematics 1979, 1-101 (2009).
This is an expository article on the elementary theory of rough paths, where a special emphasis is put on proper definitions of rough paths from an algebraic point of view, basically in line with [A. Lejay, 37th seminar on probability. Berlin: Springer. Lect.Notes Math.1832, 1–59 (2003; Zbl 1041.60051)], or see e.g. [T. Lyons and Z. Qian, System control and rough paths. Oxford Mathematical Monographs. Oxford: Clarendon Press (2002; Zbl 1029.93001)]. Starting with simple considerations on ordinary integrals and stressing the importance of the Green-Riemann formula, the author is aiming at an introduction to another point of view on the theory of rough paths, which is suggested by the work of D. Feyel and A. de La Pradelle [Electron.J.Probab. 11, Paper No. 35, 860–892, electronic only (2006; Zbl 1110.60031)], where they use a point of view from differential geometry and emphasize the importance of the Gauss/Green-Riemann/Stokes formula to understand the need to enhance the path with more information to get a rigorous definition of rough paths. Based upon this point of view, the construction of the algebraic structures such as tensor space and Lie groups, etc. is naturally justified, which is needed in the theory of rough paths from basic considerations on integrals of differential forms. The author tries to introduce, as easily and clearly as possible, alternative ways to understand why the construction of [T. Lyons et al., Lecture Notes Math.1908. Berlin: Springer (2007; Zbl 1176.60003)] is a natural generalization of the notion of integral of differential forms, in the sense that it shares the same smooth paths when one uses the right notion of a path.
More precisely, Section 2 is an introduction to basic notation and some elementary facts about integrals of differential forms along smooth paths as well as about Hölder continuous paths. Actually it includes the class of \(\gamma\)-Lipschitz differential forms \(\text{Lip}(\gamma; {\mathbb R}^d \to {\mathbb R}^m)\), the integral of a continuous differential form \(f\) along a path \(x \in\) \(C^1( [0,T]; {\mathbb R}^d)\) \[ \int_x f = \sum_{i=1}^d \int_0^T f_i(x_s) \left. \frac{ d x^i}{dt} \right|_{t=s} ds, \] the Green-Riemann/Stokes/Gauss formula, and the notion of finite \(p\)-variation. In Section 3, following a quick review on Young integrals \({\mathcal I}(x; s,t)\) \(=\) \(\int_{ x | [s,t]} f\), properties of integrals along \(\alpha\)-Hölder continuous paths with \(\alpha > 1/2\) are shown. The section ends with a practical counter-example in the stochastic setting arising from homogenization theory, which provides the cornerstone to understand how \({\mathcal I}\) will be defined so as to deal with irregular paths. In Section 4, under the assumption that one can integrate differential forms along \(\alpha\)-Hölder continuous path with \(\alpha \in\) \(( 1/3, 1/2 ]\), it is discussed in detail how to transform this integral \({\mathcal I}\) into a continuous one with respect to the path. In Section 5, the space \(A({\mathbb R}^2)\) is constructed, which is nothing but the space \({\mathbb R}^3\) equipped with operation \(\boxplus\) as a non-commutative group. Then, considering paths taking values in \(A({\mathbb R}^2)\), the author shows how to define the integral \(\int_x f\) as a limit of ordinary integrals. Analysis of the space \(A({\mathbb R}^2)\) is continued in Section 6, and the author introduces another formulation for the integral \({\mathcal I}\) for a path \(x \in\) \(C^{\alpha}([0,T]\); \(A({\mathbb R}^2))\). The tensor space \(T({\mathbb R}^2)\) \(=\) \({\mathbb R} \oplus {\mathbb R}^2\) \(\oplus\) \(( {\mathbb R}^2 \otimes {\mathbb R}^2)\) and its subspaces \[ T_{\xi} ({\mathbb R}^2 ) = \{ ( \xi, x^1, x^2) | \, x^1 \in {\mathbb R}^2, x^2 \in {\mathbb R}^2 \otimes {\mathbb R}^2 \} \quad \text{with} \quad \xi \in \{ 0,1\} \] are introduced, and the Riemannian structure on \(T_1( {\mathbb R}^2)\) is also investigated. Section 7 is devoted to rough paths and their integrals. As a matter of fact, a rough path \(x \in\) \(C^{\alpha}([0,T]\); \(T_1({\mathbb R}^2))\) and a geometric rough path are defined. Another integral \({\mathcal I}(x)\) of \(f\) along \(x \in\) \(V^{\alpha}([0,T]\); \(T_1({\mathbb R}^2))\) is constructed, using an expression of Riemann sum type. This construction just corresponds to the original one of T. Lyons [Rev.Mat.Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)]. Some related results are given in Section 8. For example, the case of a path living in a \(d\)-dimensional space, Chen series using iterated integrals, other constructions for paths with quadratic variation, and some link with stochastic integrals are discussed. Section 9 treats the differential equation \[ y_t = y_0 + \int_0^t g(y_s) d x_s \] with an irregular path \(x\), where the theory of rough paths are applied to solve the differential equation. This article ends with an appendix which includes
Carnot groups and homogeneous gauges and norms,
Brownian motion on the Heisenberg group, and
From almost rough paths to rough paths.
This article is well written almost in self-contained manner; it succeeds in making the rough path theory (a really touchy issue) easily accessible even to non-expert readers.
For the entire collection see [Zbl 1166.60002].


60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60Hxx Stochastic analysis
60H05 Stochastic integrals
58A10 Differential forms in global analysis
Full Text: DOI


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