Related papers: Rough integrators on Banach manifolds
By using an explicit ordinary differential equation to approximate the exponential solution flow, we extend the universal limit theorem to rough differential equation in Banach space driven by weak geometric rough path, and give the…
Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one…
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $\alpha\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly…
We introduce a differential structure for the space of weakly geometric p rough paths over a Banach space V for 2<p<3. We begin by considering a certain natural family of smooth rough paths and differentiating in the truncated tensor…
This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric \Pi-rough paths in our terminology) sketched by Lyons ("Differential equations driven by rough signals", Revista Mathematica Iber. Vol 14, Nr.…
This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We…
Using truncated variation techniques we obtain an improved version of the Loeve-Young inequality for the Riemann-Stieltjes integrals driven by rough paths. This allowed us to strenghten some result on the existence of solutions of integral…
This paper establishes the existence and uniqueness of solutions for rough differential equations driven by reduced rough paths with low regularity, specifically in the roughness regime $\frac{1}{3} < \alpha \leq \frac{1}{2}$. While the…
We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-gamma manifolds.
We provide a draft of a theory of geometric integration of rough differential forms which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving H\"older continuous…
We embed the rough integration in a larger geometrical/algebraic framework of integrating one-forms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated…
When the one-form is $Lip\left(\gamma-1\right) $ with $\gamma >p\geq 1$, we construct the integral of a branched $p$-rough path, which defines another branched $p$-rough path. We derive a quantitative bound for this integral and prove that…
The concept of Rademacher type $p$ ($1\leq p\leq2$) plays an important role in the local theory of Banach spaces. In \cite{mas88} Mascioni considers a weakening of this concept and shows that for a Banach space $X$ weak Rademacher type $p$…
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…
We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that…
We prove some results, which are used in arXiv:1406.7871, about weakly geometric rough paths that are well-known in finite dimensions, but need proof in the infinite dimensional setting.
We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We…
The paper is devoted to obtaining conditions for the roughness of dichotomy in the Banach spaces. Deep analysis of the well known papers was considered. The main results also works for the case of unbounded operators.
We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of…
In this paper, we introduce a new class of subsets of bounded linear operators between Banach spaces which is p-version of the uniformly completely continuous sets. Then, we study the relationship between these sets with the equicompact…