Related papers: A uniform estimate for rough paths
By studying $\mathbb{A}^1$-curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete…
A generalized divergence theorem is established allowing for domains with inner boundaries. The normal trace of a rough integrand is not a Radon measure; rather, the boundary integral is expressed via a surface functional continuous with…
In this paper, we study two variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes…
We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph $G$ with two distinct terminal vertices and two positive integers $p$ and $k$, the question is…
Motivated by recent applications in rough volatility and regularity structures, notably the notion of singular modelled distribution, we study paths, rough paths and related objects with a quantified singularity at zero. In a pure path…
We establish sharp estimates for the convergence rate of the Kranosel'ski\v{\i}-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of…
In line with the notion of probabilistic rough paths introduced in the previous contribution \cite{salkeld2021Probabilistic}, we address corresponding random controlled rough paths (first introduced in \cite{2019arXiv180205882.2B}), the…
The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and…
We present a new analytical method that systematically improves the convergence of path integrals of a generic $N$-fold discretized theory. Using it we calculate the effective actions $S^{(p)}$ for $p\le 9$ which lead to the same continuum…
The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the D\"urr-Connes-Kreimer…
Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…
Inspired by recent advances in singular SPDE theory, we use the Poincar\'e inequality on Wiener space to show that controlled complementary Young regularity is sufficient to obtain Gaussian rough paths lifts. This allows us to completely…
We present a few techniques for proving $L^p$ estimates for martingales. Basic applications to It\^o integration and rough paths are included.
We establish an uniform factorial decay estimate for the Taylor approximation of solutions to controlled differential equations. Its proof requires a factorial decay estimate for controlled paths which is interesting in its own right.
We give a classification and a construction of all smooth $(n-1)$-dimensional varieties of lines in ${\bf P}\sp n$ verifying that all their lines meet a curve. This also gives a complete classification of $(n-1)$-scrolls over a curve…
We consider the oscillatory integrals with parameter-dependent phases. We decompose the integrals into a leading term and a remainder term. Instead of the pointwise estimate, we use some $L^p$-estimate for the remainder term and get various…
We prove a generalisation of Fernique's theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough path theory where one deals with iterated…
In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of…
In this complementary note to [1] (arXiv:1501.05641), we provide an alternative proof for the factorial decay estimate of iterated integrals for geometric rough paths without using the neoclassical inequality. This note intends to aid the…
Optimal sample path properties of stochastic processes often involve generalized H\"{o}lder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $\psi (x) \equiv $…