Related papers: A uniform estimate for rough paths

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 consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative…

Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…

We prove the existence and uniqueness of solutions to a class of stochastic scalar conservation laws with joint space-time transport noise and affine-linear noise driven by a geometric p-rough path. In particular, stability of the solutions…

In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by a epsilon-periodic lattice of highly conducting thin rods. The estimates are derived…

We prove sharp homogeneous improvements to $L^1$ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These…

The signature of a $p$-weakly geometric rough path summarises a path up to a generalised notion of reparameterisation. The quotient space of equivalence classes on which the signature is constant yields unparameterised path space. The study…

We make use of the complex implicit representation in order to provide a deterministic algorithm for checking whether or not two implicit algebraic curves are related by a similarity, a central question in Pattern Recognition and Computer…

A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H(W)+t_H(1-W)\geq 2^{1-e(H)}$ holds for…

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with…

Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the…

Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, $E$ a separable real Banach space and $p\geq 1$. Given a sequence of functions $f, f_1, f_2,...$ from $T\times E$ to ${\bf R}$, under general assumptions, we prove that, for each…

We give a new proof of the trace formula for regular graphs. Our approach is inspired by path integral approach in quantum mechanics, and calculations are mostly combinatorial.

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to the fractional…

We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these…

We provide $L^1$ estimates for a class of transport equations containing singular integral operators. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is…

Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the…

Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio…

The main purpose of this paper is to investigate the strong approximation of the $p$-fold integrated empirical process, $p$ being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of…