arXiv research

The paper introduces surface signatures for irregular surfaces and rough surfaces.

Discretizes diffusions and harmonic functions on covering spaces.

For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory.

We show that if $p \colon M \to N$ is a normal Riemannian covering, with $N$ closed, and $M$ has exponential volume growth, then there are non-constant, positive harmonic functions on $M$. This was conjectured by Lyons and Sullivan in \cite{LS}.

Paper proves GDL models can approximate any continuous function on non-Euclidean data.

New algorithm reduces regret and constraint violation in constrained bandit problems.

Improved bounds on neural network expressivity.

Unified approach to stochastic control, filtering, and stopping using rough paths.

Lyons and Sullivan have shown how to discretize harmonic functions on a Riemannian manifold $M$ whose Brownian motion satisfies a certain recurrence property called $\ast$-recurrence. We study analogues of this discretization for tensor fields which are harmonic in the sense of the covariant Laplacian. We show that, un…

Cubature on Wiener space [Lyons, T.; Victoir, N.; Proc. R. Soc. Lond. A 8 January 2004 vol. 460 no. 2041 169-198] provides a powerful alternative to Monte Carlo simulation for the integration of certain functionals on Wiener space. More specifically, and in the language of mathematical finance, cubature allows for fast…

High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical me…

Market events such as order placement and order cancellation are examples of the complex and substantial flow of data that surrounds a modern financial engineer. New mathematical techniques, developed to describe the interactions of complex oscillatory systems (known as the theory of rough paths) provides new tools for…

New relation on paths is not transitive.

Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assum…

A new proof of an extension theorem with bounded generators.

Paper develops a high-order recombination algorithm for financial modeling.

Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka~[Adv.~Math.~Econ.~6, 69--83, 2004] and Lyons--Victoir~[Proc.~R.~Soc.\\Lond.~Ser.~A 460, 169--198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the Ninomiya-Victoir algorithm~[Appl.~Math.~Fin.~15, 1…

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…

Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.

Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.

We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem

Paper extends theorem on covering spaces and Jordan curves.

In this paper, we study the general extension problem for isometric immersions by establishing Cartan-Ambrose-Hicks theorems based on submanifolds. Our method also provides geometric constructions of such extensions.

We give a complementary generalization of the extensions of Bonnet-Myers theorem obtained by Calabi and also Cheeger-Gromov-Taylor.

The abstract presents a new theorem using Ross-Witt Nyström correspondence and Berndtsson's theorem.

Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.

We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.

We prove an extension theorem for Kahler currents with analytic singularities in a Kahler class on a complex submanifold of a compact Kahler manifold.

New Klein-Maskit theorems for Anosov subgroups.

Quadratic differentials on Riemann surfaces uniquely determine foliations.

New extension theorem for projective manifolds.

In this article, we prove a Kahler extension theorem for real Kahler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kahler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states th…

In this paper, we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature.

A new method for portfolio optimization using signature signatures to incorporate path-dependencies.

Paper extends Ohsawa-Takegoshi theorem to more general domains, proving removable singularities for plurisubharmonic functions.

Extends Seeley's theorem for Bastiani's differential calculus in infinite dimensions.

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answ…

In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension $d\geq1$, which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to Le-Šešum \cite{LS} and the authors \cite{XYZ1,XYZ2}. Usin…

The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.

The paper proves extension theorems for complex manifolds with Levi $q$-concave domains.

Kernel for Lévy rough paths derived from PDE system.

A statistical test of independence may be constructed using the Hilbert-Schmidt Independence Criterion (HSIC) as a test statistic. The HSIC is defined as the distance between the embedding of the joint distribution, and the embedding of the product of the marginals, in a Reproducing Kernel Hilbert Space (RKHS). It has …

In this paper, we prove the extensions of Bonnet--Myers' type theorems obtained by Calabi and Cheeger--Gromov--Taylor via Bakry--Emery Ricci curvature, which generalize the results of \cite{FG, Lim1, Wan, Wang, WW, Wu}.

The paper defines positivity for singular metrics on vector bundles and proves related theorems.

We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 199…

This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.

The proof of Theorem 7.12 of "Uniqueness of smooth cohomology theories" by the authors of this note is not correct. The said theorem identifies the flat part of a differential extension of a generalized cohomology theory E with ER/Z (there called "smooth extension"). In this note, we give a correct proof. Moreover, we …