• Corpus ID: 247411237

The geometry of controlled rough paths

  title={The geometry of controlled rough paths},
  author={Mazyar Ghani Varzaneh and Sebastian Riedel and Alexander Schmeding and Nikolas Tapia},
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… 

Introduction to rough paths theory

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



A selection theorem for Banach bundles and applications

  • A. Lazar
  • Mathematics
    Journal of Mathematical Analysis and Applications
  • 2018

A theory of regularity structures

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

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

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

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

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

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

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