Perturbed linear rough differential equations

Abstract

We study linear rough differential equations and we solve perturbed linear rough differential equation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebra more general than tensor algebra and to consider extensions of classical results such as the Magnus and the Chen-Strichartz formula. IntroductionLinear Rough Differential Equations (RDE) have been considered by several authors since they are an essential tool for studying the derivative of the Itô map and its flow properties (See the bibliography in [21]). However, with the exception of the works of D. Feyel, A. de la Pradelle and G. Mokobodzki [27] and S. Aida [1], linear RDE have hardly been considered as objects as such with their own properties, excepted to control the growth of the solution as in [30,40]. Instead, they are generally presented as a special case of RDE. However, the Baker-Campbell-Hausdorff-Dynkin formula was among the inspirations of the theory of rough paths [4,9]. The algebraic view of solution of controlled differential equations through for example Chen-Fliess series * This work has been supported by the ANR Ecru (ANR-09-BLAN-0114-01/02). A. Lejay also want to thank the Centre IntraFacultaire Bernoulli (CIB) and the NSF (National Science Foundation) for its kind hospitality during the SPDE Semester in 2012.

Sensitivity of rough differential equations: An approach through the Omega lemma

“…Since Y satisfies Y r,s Y s,t = Y r,t , y satisfies (14). From (17), we easily obtain (15) and (16).…”

“…We provide a "genuine rough path" approach which removes the restriction implied by smooth rough paths, the restriction to geometric rough paths (which could be dealt otherwise with (p, p/2)-rough paths [41]) as well as any restriction on the dimensions of the Banach spaces U and V. Although we use a CRP, the Duhamel formula shown in [17] could serve to prove a similar result for (partial) rough paths and not only CRP.…”

An Itô Formula for rough partial differential equations and some applications

“…The previous discussion depicts a non-commutative generalization of the usual rough paths theory, which has been already discussed e.g. in [4,5,13,21]. In this picture, real numbers -in which the coordinates of a path Z : [0, T ] → R m live -are substituted by elements of an algebra (here a space of differential operators), and the constraint (2.7) corresponds to Chen's relations.…”

The non-linear sewing lemma I: weak formulation

“…of [16] and of [12], introduced to solve linear RDE to a non linear situation. In this way, we construct directly some flows.…”

“…This theory also found applications in machine learning and the recognizing of the Chinese ideograms [10,24].Since the seminal article [25] by T. Lyons in 1998, several approaches emerged to solve (1). They are based on two main technical arguments: fixed point theorems [20,25] and flow approximations [2,12,14,16,19]. In particular, the rough flow theory allows one to extend work about stochastic flows, which has been developed in '80s by Le Jan-Watanabe-Kunita and others, to a non-semimartinagle setting [4].The main goal of this article is to give a framework which unifies the approaches by flow and pursue further investigations on their properties and their relations with families of solutions to (1).A flow is a family of maps t , u from a Banach space to itself such that ,˝, " , for any ď ď .…”