Flows Driven by Banach Space-Valued Rough Paths

Bailleul, Ismaël

doi:10.1007/978-3-319-11970-0_7

Cited by 29 publications

(46 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The 4-points control can be checked on the almost flow, which is then called a stable almost flow. This condition is weaker that the one given by I. Bailleul in [2,5]: there it should roughly be C 1 with a Lipschitz continuous spatial derivative while in our case, the spatial derivative may be only Hölder continuous. The question of the existence of a Lipschitz flow without the UL condition, in relation with Stochastic Differential Equations, should be dealt with in a subsequent work.…”

Section: Introductionmentioning

confidence: 62%

“…Finally, we apply our framework to recover the results of A.M. Davie [14], P. Friz & N. Victoir [17,19] and I. Bailleul [2,5] using various perturbation arguments. Although not done here, our framework could be applied to deal with branched rough paths, that are high-order expansions indiced by trees, which are studied in [9] and shown to fit the Bailleul's framework [3].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The non-linear sewing lemma I: weak formulation

Brault¹,

Lejay²

2019

Electron. J. Probab.

View full text Add to dashboard Cite

We give an unified framework to solve rough differential equations. Based on flows, our approach unifies the former ones developed by Davie, Friz-Victoir and Bailleul. The main idea is to build a flow from the iterated product of an almost flow which can be viewed as a good approximation of the solution at small time. In this second article, we give some tractable conditions under which the limit flow is Lipschitz continuous and its links with uniqueness of solutions of rough differential equations. We also give perturbation formulas on almost flows which link the former constructions.This theory was very fruitful to study stochastic equations driven by Gaussian process which is not covered by the Itô framework, like the fractional Brownian motion [13,26]. More generally, the rough path framework allows one to separate the probabilistic from the deterministic part in such equation and to overcome some probabilistic conditions such as using adapted or non-anticipative processes.Recently, the ideas of the rough path theory were extended to stochastic partial differential equations (SPDE) with the works of [21,22] which have led to significant progress in the study of some SPDE. 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 . Typically, the map which associates the initial condition to the solution of (1) has a flow property. The existence of a such flow heavily depends on the existence and uniqueness of the solution. However, it was proved in [7,8] and extended to the rough path case in [6] that when non-uniqueness holds, it is possible to build a measurable flow by a selection technique. In this article we are interested by the construction of a Lipschitz flows.The main idea to build the flow associated is to find a good approximation , of , when |´| is small enough. We iterate this approximation on a subdivision " t ď ď¨¨¨ď ď u of r , s by setting , :" ,˝¨¨¨˝, . Ifdoes converges when the mesh of goes to zero, , the limit is necessarily a flow This computation is similar to the ones of numerical schemes as Euler's methods of different order [11]. Moreover, this idea is found among the Trotter's formulas for bounded or unbounded linear operators which allows to compute the semi-group of the sum of two non-commutative operators only knowing the semi-groups associated to e...

show abstract

Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

The non-linear sewing lemma I: weak formulation

Brault¹,

Lejay²

2019

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…for all t close enough to s for x t to belong to V s , and any function f of class C [p]+1 defined on V s , where it has bounded derivatives. The results of [16] show that such a rough differential equation has a unique maximal solution started from any given point, as awaited. It also follows from the results of [16], or other classical works, that the solution path x • depends continuously on the driving signal X in the following sense.…”

Section: Rough Integrators On Banach Manifoldsmentioning

confidence: 88%

Rough integrators on Banach manifolds

Bailleul

2019

Bulletin des Sciences Mathématiques

View full text Add to dashboard Cite

show abstract

“…Our framework is close to the one developed by I. Bailleul in [2,5] as we also give two conditions which ensure that the iterated composition of the almost flows remain uniformly Lipschitz continuous, which we called in [7] the UL condition.…”

Section: Introductionmentioning

confidence: 85%

The non-linear sewing lemma II: Lipschitz continuous formulation

Brault

Lejay

2021

Journal of Differential Equations

View full text Add to dashboard Cite

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

show abstract

Flows Driven by Banach Space-Valued Rough Paths

Cited by 29 publications

References 20 publications

The non-linear sewing lemma I: weak formulation

The non-linear sewing lemma I: weak formulation

Rough integrators on Banach manifolds

The non-linear sewing lemma II: Lipschitz continuous formulation

Contact Info

Product

Resources

About