arXiv research

A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodes…

Complete Finsler spaces with negative Ricci curvature are reversible.

Established a Hardy inequality on Finsler manifolds.

We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We …

In 3D space forms, a lens minimizes volume for a fixed surface area.

We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usual Holder inequality, which may be of independent interest.

New inequality for eigenfunctions on curved spaces.

The paper analyzes variance reduction in stochastic gradient Langevin dynamics.

We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.

The paper solves reverse isoperimetric problems for convex bodies with curvature constraints.

Paper optimizes diffusion models for denoising tasks with theoretical guarantees.

In this paper, we study the sharp constants of quantitative Hardy and Rellich inequalities on nonreversible Finsler manifolds equipped with arbitrary measures. In particular, these inequalities can be globally refined by adding remainder terms like the Brezis-Vázquez improvement, if Finsler manifolds are of strictly ne…

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by opti…

A classical result by Alexander Grigor'yan states that on a stochastically complete manifold the non-negative superharmonic $L^1$-functions are necessarily constant. In this paper we address the question of whether and to what extent the reverse implication holds.

We address the problem of estimating the mixing time $t_{\mathsf{mix}}$ of an arbitrary ergodic finite-state Markov chain from a single trajectory of length $m$. The reversible case was addressed by Hsu et al. [2019], who left the general case as an open problem. In the reversible case, the analysis is greatly facilita…

The paper proves inequalities for Steklov eigenvalues on finite graphs.

In financial markets, low prices are generally associated with high volatilities and vice-versa, this well known stylized fact usually being referred to as leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts of leverage a…

DDS samples from noisy data by reversing diffusion, providing theoretical guarantees.

Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on…

In this paper we study the Finsler sphere $(S^n,F)$ with $n>1$, which has constant flag curvature $K\equiv 1$ and only finite prime closed geodesics. In this case, the connected isometry group $I_0(S^n,F)$ must be a torus which dimension satisfies $0<\dim I(S^n,F) \leq[\frac{n+1}{2}]$. We will prove that the number of …

PDDS samples from unnormalized densities using iterative particle scheme.

Study counts and equidistributes geodesic orbits on curved spaces.

We consider an SPDE description of a large portfolio limit model where the underlying asset prices evolve according to certain stochastic volatility models with default upon hitting a lower barrier. The asset prices and their volatilities are correlated via systemic Brownian motions, and the resulting SPDE is defined o…

The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.

We prove that the length of the boundary of a $J$-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the rea…

The study proves properties of optimizers for sets maximizing perimeter under fixed volume constraints.

Classifies reversible and strongly reversible elements in quaternionic groups.

The paper classifies reversible and strongly reversible elements in Hermitian isometry groups.

A new model predicts price concavity and reversion after metaorder execution.

This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (θ+σα(R_t, t))R_t dt +σR_t dB_t \] with an initial value $R_0=r_0\in\mathbb{R}$, where $θ\in\mathbb{R}$ and $σ>0$ are constants, and the mean correction function $α:\mathbb{R}\times[0,\infty)\to α(x,t)\…

In electricity markets, it is sensible to use a two-factor model with mean reversion for spot prices. One of the factors is an Ornstein-Uhlenbeck (OU) process driven by a Brownian motion and accounts for the small variations. The other factor is an OU process driven by a pure jump Lévy process and models the characteri…

This paper classifies reversible and strongly reversible elements in affine groups.

The paper is devoted to weighted $L^p$-Hardy inequalities with best constants on Finsler metric measure manifolds. There are two major ingredients. The first, which is the main part of this paper, is the Hardy inequalities concerned with distance functions in the Finsler setting. In this case, we find that besides the …

New model generates images by reversing heat equation, revealing disentanglement.

The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get semi-local variants of the inequality holding in any dimension that involve domai…

New neural net learns time-reversible symplectic dynamics.

A new trading strategy using reinforcement learning for statistical arbitrage.

Algebraic method reveals criterion for quaternionic Möbius group reversibility.

Recent studies have shown that online portfolio selection strategies that exploit the mean reversion property can achieve excess return from equity markets. This paper empirically investigates the performance of state-of-the-art mean reversion strategies on real market data. The aims of the study are twofold. The first…

Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian…

Sharp stability results for reverse isoperimetric inequalities in 2D.

In this paper, we introduce metallic maps between metallic Riemannian manifolds, provide an example and obtain certain conditions for such maps to be totally geodesic. We also give a sufficient condition for a map between metallic Riemannian manifolds to be harmonic map. Then we investigate the constancy of certain map…

Let $G$ be a group. An element $g$ in $G$ is called reversible if it is conjugate to $g^{-1}$ within $G$, and called strongly reversible if it is conjugate to its inverse by an order two element of $G$. Let $\textbf{H}_{\mathbb H}^n$ be the $n$-dimensional quaternionic hyperbolic space. Let $\mathrm{PSp}(n,1)$ be the i…

It is the Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) distance functions on a bounded convex domain in R^n such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective.…

On-line portfolio selection has attracted increasing interests in machine learning and AI communities recently. Empirical evidences show that stock's high and low prices are temporary and stock price relatives are likely to follow the mean reversion phenomenon. While the existing mean reversion strategies are shown to …

Boundary rigidity proven for non-reversible Finsler metrics.

Conditional diffusion models can approximate target distributions well with Gaussian-mixture reverse kernels.