# The geometry of controlled rough paths

@inproceedings{Varzaneh2022TheGO, title={The geometry of controlled rough paths}, author={Mazyar Ghani Varzaneh and Sebastian Riedel and Alexander Schmeding and Nikolas Tapia}, year={2022} }

We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the…

## One Citation

### Introduction to rough paths theory

- Computer Science
- 2023

These notes are an extended version of the course ``Introduction to rough paths theory'' given at the XXV Brazilian School of Probability in Campinas in August 2022. Their aim is to give a consise…

## References

SHOWING 1-10 OF 36 REFERENCES

### A selection theorem for Banach bundles and applications

- MathematicsJournal of Mathematical Analysis and Applications
- 2018

### A theory of regularity structures

- Mathematics
- 2014

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each…

### Oseledets Splitting and Invariant Manifolds on Fields of Banach Spaces

- MathematicsJournal of Dynamics and Differential Equations
- 2019

We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove…

### Character groups of Hopf algebras as infinite-dimensional Lie groups

- Mathematics
- 2015

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie…

### Geometric versus non-geometric rough paths

- Mathematics
- 2012

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…

### On the Lie envelopping algebra of a pre-Lie algebra

- Mathematics
- 2004

We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. Then we proove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We…

### Pre-Lie algebras and the rooted trees operad

- Mathematics
- 2000

A Pre-Lie algebra is a vector space L endowed with a bilinear product * : L \times L to L satisfying the relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), for all x,y,z in L. We give an explicit…

### Combinatorics of rooted trees and Hopf algebras

- Mathematics
- 2003

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the…